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I 



The 
Ancient 
Tradition 
of Geometric 
Problems 


Wilbur Richard Knorr 


Included wilhin ilie medical encyclopedia of Dioscornlcs <11. mid-first century 
A.D.) is n survey of the plant species and their therapeutic value. Ilie drawings 
of ivy shown here are taken from the famous illustrated Uy/antine manuscript of 
the work (made in 512 A.D.) and now held in the collection of ancient manu¬ 
scripts of the Austrian National Library (Cod. Vimlobon. Med. Gr. 1. f. I74v). 
This reproduction has been made from the facsimile edition of the Codex (Aka- 
dcmischc Dntck- und Verlagnnsinlt. Graz. 1966-70, vol, 2), with the kind per¬ 
mission of the publishers and ihe Ocstcrrcicliische Nutionnlbihliothck. (Lor dis¬ 
cussion. see chap. 6. sect, iv.) 


Birkhauser 


Boston • Basel • Stuttgart 













VI 


Table <>l Cements 


5 Archimedes—The Perfect Eudoxcan Geumeter 151 

(i) Circle-Quadrature and Spirals 153 

(ii) Problem*Solving via Conic Seel ions 170 

(iii) Problem-Solving via Ntuxes I7K 

(iv) An Ammynww* Cubc*IX»plicaiion IKK 

(v) 11* Impact of Arcliinictlcs* Wink 194 

6 The Successors of Archimedes in the 3ul Century 209 

(i) EratoMlicnes 210 

(ii) Nicomcdes 219 

(iii) Diocles 233 

(iv) On the Curve called "Cissoiil" 246 

(v) Oionysodtmis, Perseus and Zenotloius 263 

(vi) In the Shadow of Archimedes 274 

7 Apollonius—Culmination of the Tradition 293 

(i) Apollonius, Archimedes and llcraclides 294 

(ii) Apollonius and Nicomcdes 303 

(iii) Apollonius and Uuclid 313 

(iv) Apollonius and Aristaeus 321 

(v) Origins ami Motives of the Apollonian Oeomeliy 32K 

8 Appraisal of Ihe Analytic Field in Antiquity 339 

(i) The Ancient Classiliealions of Problems 341 

(ii) Problems, Theorems and the Method of Analysis 34K 

(iii) •*... and many and the greatest sought, but did not find.” 361 

(iv) Hpdoguc 367 

Bibliography 382 


Indices 393 


Preface 


Within the ancient geometry, a geometric "problem” seeks the construction of 
a figure corresponding It* a specific description. The solution to any problem 
requires for its completion an appeal to the constructions in other problems 
already solved, anil in turn will be applied to the solutions of yet others. In 
effect, then, the corpus of solved problems forms an ordered sequence ill which 
each problem can he reduced to those preceding. The implications of this simple 
conception struck me, as I was completing a paper tin Apollonius’ construction 
of the hyperbola (1980; published in Cenumrus, 19K2), for it served to unify a 
diverse range of geometric materials I bad then l>ccn collecting lor some live 
years. 

’dial the ancient problem-solving effort look on such a structure is much what 
one would expect, given the prominence of the role of “analysis” for discovering 
and proving solutions; for this method seeks in each ease to reduce the slated 
problem to others already solved. We possess one ancient work, tutelar s Data, 
which organizes the materials of elementary geo me try in (his manner; but there 
survives none which attempts the same lor ihe more advanced field. 'Hie Conics 
of Apollonius, lor instance, is by its own account ancillary to this effort, pro¬ 
viding the essential introduction to the theory of conics through which the so- 
called "solid” problems might he solved; Init it undertakes the actual solution 
of such problems only to a very limited extent. This salient omission from the 
extant record thus defines the project of the present work: to exploit Ihe materials 
extant from Archimedes, Apollonius, Pappus and others in order to retrieve a 
sense of the nature and development of this ancient tradition of analysis. 

The present cffoil is conceived as an exploratory essay, intended to reveal 
the opportunities which the evidence available to us provides for an interpretation 
of the ancient field. A definitive survey overreaches its scope, however, and 
may ultimately be unattainable in view of the gaps in our documentation. Sim- 





Preface 


u 



viit 

ilarly, an exhaustive survey of the immense secondary literature on the history 
of geometric constructions could not he attempted here. The sheer hulk of this 
literature has made omissions inevitable, but I have endeavored to include ref¬ 
erences to those contributions which seemed to me historically and technically 
stimulating, as well as directly pertinent to my special objectives. If through 
inadvertence I have omitted discussions of worth, I apologize for that and wel¬ 
come information which wilt prevent similar lapses in the future. 

Since (be field of ancient geometry, in particular, the work of Archimedes, 
has been an area of special interest to me for several years, (he present study 
occasionally touches on issues discussed in previous publications of my own. 
Thus, I sometimes have reason to summarize positions elaborated in detail else¬ 
where, or to allude in passing to treatments of related issues, in such eases I 
have striven to avoid mere repetition of previous efforts, and the present work, 
but for these occasional overlaps, consists of new material. 

In my interpretive efforts here I will emphasize the mathematical and historical 
aspects of the ancient writings, while the final chapter will include aspects of 
philosophical interest as well. Another aspect, that dealing with textual issues 
of documentation, while often overlooked in studies of this type, sometimes 
becomes indispensable for understanding the ancient work. I thus include com¬ 
mentary along these lines, when it bears on specific matters taken up here. The 
broader inquiry, of surveying the ancient documents and working out an account 
of their production and transmission, is not attempted here, but forms the project 
of a separate volume currently in preparation. I intend to examine in that second 
work a set of Greek and Arabic texts on geometric problems, thereby to reveal 
the general character of the scholarship which has preserved testimony of the 
ancient geometric field. 

Grants from the American Council of learned Societies and the Institute for 
Advanced Study enabled me to undertake these researches at the Institute in 
1978-79, and a grant from the National Science Foundation, administered by 
the American Academy of Ails and Sciences, provided support for continuation 
of this work in 1979-80, 

1 have profited considerably from the criticisms and suggestions of students 
in the Department of Philosophy at Stanford University, in particular, Susan 
Hollander, Henry Mcndcll and David O’Connor. I wish also to thank my col¬ 
leagues in the Departments of Philosophy, Classics and Mathematics for their 
comments, and, in particular, Sol Fefertnun, Halsey Koyden, Hans Samclson 
and Patrick Suppes, for reading draft chapters, and Jody Maxmin for advice in 
the presentation of the plates, Ian Mueller (University of Chicago) provided 
extensive and valuable comments on the work in its later stages. I especially 
appreciate the respect and courtesy with which they all received my effort, even 
when my positions differed sharply from their own. 

I am cndebicd to a number of museums and research libraries for permission 
to reproduce photographs of items in their collections. 'Riese are noted in the 
captions to the frontispiece and ihe plates. A grant from Stanford University, 


through the Office of its Dean of Graduate Study and Research, has contributed 
toward the costs of publishing these photographic materials. 

Finally, l wish to acknowledge the support, patience and diligence of the 
president and editors of Rirkhauscr Boston, Inc. 

W.R.K. 
Stanford, Calif. 




CHAPTER 


1 


Sifting 

History 

from 

Legend 


Hie problems til cube duplication, angle trisect ion, and circle quadrature have 
deservedly attracted major interest among scholars of ancient (ireek geometry. 
The ancient literature on these problems constitutes tine of the most richly doc¬ 
umented fields of ancient mathematics, amounting lo over two dozen different 
solutions, often in multiple versions and spanning the whole course ol antiquity 
from the prc-Hudidcan period through the Hellenistic, Graeco-Roman, and By¬ 
zantine periods even into the Arabic Middle Ages. Throughout, these problems 
were affiliated with researches in the more advanced fields of geometry. 

A historical survey of these studies could thus be hoped to provide insight of 
value into the development of the techniques of geometry in antiquity: What 
were the sources of the interest in these problems*' How did their study relate 
to prior efforts on the same or associated problems? How did the results of these 
studies contribute to the development of new geometric techniques for the so¬ 
lution of geometric problems? What were the precise specifications for the con¬ 
struction of solutions, and did these conditions themselves change with the 
introduction of new techniques and concepts? Ultimately, did the ancient geo¬ 
meters and philosophers view the quest lor solutions as having succeeded? 

Unfortunately, only a small part of this project now permits its presentation 
in the form of a straightforward narrative of how things happened. The extant 
technical literature has large gaps, so that we cannot claim to know all the 
solutions to these problems which the ancients worked out, nor even, in some 
cases, the gcomctcis responsible for solutions which have survived. More dif¬ 
ficult still is the assessment of motive, since technical documents rarely provide 





2 Ancient Tradition «l Gcoiiicliii Problems ' 

direct insight into the reasons which lead geometers to take up specific problems 
liitd treat them in specific ways, Por this we arc often compelled to resort to ; 

testimonies in nonmathematical writings. But we then face new difficulties, in ! 

|hat their authors may not fully comprehend the technical issues and. at any rate. 

[will have their own literary or philosophical concerns. The latter will invariably 
discourage them from presenting the technical materials in the clem and detailed 
manner we would prefer. 

ly One of the important witnesses to the early study of the cube duplication j 

[illustrates well the hazards of using such nonmalhcniatical sources. It is a passage j 

from the Demon of Socrates, a dramatization of the political conspiracies in the ? 

Greek city-state of Thebes in 379 nr by a writer best known for his philosophical i 

and biographical works. Plutarch of Cluieronea (1st-2nd century a.d ,). 1 The * 

speaker here is Simmias. one of the Theban conspirators, who is reporting on 
la recent visit to Egypt he made in the company of Plato: 

£ As we traveled from Egypt, certain Delians came upon us near t aria ami begged 

* Plato, as a geometer, to solve for them an unusual oracle proposed by the god. 
t The oracle was that there would l>c a respite from present ills for the Delians and 
f the other Greeks after they had doubled the altar in Delos. But as they couldn't 
it figure out the purpose and were faring ludicrously with the construction of the attar 
f? (for on account of their unfamiliarily with the proportion which provides the double 
•l in length, they failed to notice that when each of the four |!) sides was doubled. 

:* they were effecting an eightfold increase in the solid space), they summoned Plato 

for assistance in die puzzle. And he. recalling the Egyptian, said that the god was 

• making sport with the Greeks for their neglect of education, as it were taunting 
> us for our ignorance and demanding that we engage in geometry, and not just as 

•; a pastime. Por surely it is a task not for an intellect which is inferior and perceives . 

dully, but rather for one trained to the limit in (the constructions of| lines to lake 
?! two means in proportion, by which alone the figure of a cubic body is douhled hy 
being increased in the same way from every dimension. This |hc said| Eudoxus 
?; of Cnidus or Helicon ofCyzicus would work out lor them; hut they shouldn't think 
$ that the gird wished this, but rather that he enjoined all the Greeks to abandon war 
; and its ills and to consort with the Muses, and hy soothing the passions through 
reasoning and mathematics to live together profitably and withoui harm/ 

Plutarch’s chief interest here is to make the point that theoretical disciplines like 
! mathematics and philosophy can be an effective diversion from conflict, both 
•through their moderating influence on human aggressiveness and through the 
’power that reasoning has for reconciling differences. This very point had been 
;made earlier in a passage alluded to here hy the phrase “recalling the Egyptian.” 

’ in which an Egyptian priest had informed certain Greeks that the dttcument they 
^brought to him for deciphering contained an appeal to cultivate the Muses instead 
.of war. 

(- But what docs Plutarch mean to say about the origins of ihis mathematical 
inquiry itself? At first glance, we might suppose he has in mind simply that the 
geometers in Plato's circle began to study the cube duplication iti answer to the 
puzzle posed hy the oracle. In effect. Plutarch would be subscribing to a form 


.Silling llisloiy lititli Legend } 

of the “externalist" view under which technical activities find their motivation 
in factors outside the technical field, for instance, from political or economic 
causes, or. as here, from philosophical and religious ones. Indeed, passages like 
this have been cited by those who would maintain that ritual practices formed 
an important stimulus for early mathematical studies or that philosophical dis¬ 
course provided the first model of abstract reasoning for the Greek geometers.' 
But a closer reading reveals that Plutarch docs not actually claim this, for his 
Plato already knows certain facts about this problem, specifically, that it is 
equivalent to the problem of finding Ihc two mean proportionals. If the side of 
the given cube is A, and we have found two lines X. Y such that 
A : X ~ X : Y - Y : 2A. then by compounding ratios, we have 
(A : X)' = (A : X) (X : Y) (Y : 2A), whence A* : X 1 - A : 2AorX l = 2A\ 4 
Prom other sources, in particular Eratosthenes of Cyrene (3rd century n.c.h we 
learn that this reduction of the cube duplication to the finding of two mean 
proportionals was the discovery of I lippocralcs of ('bios, that is, around the turn 
of the 4lli century in: and thus predated by several decades the events on which 
Plutarch’s narrative is based.' Plutarch himself appears to depend on Eratosthenes 
for his account of the Delian oracle, for a parallel passage on the cube duplication 
in the Mathematical Exposition by Thcon of Smyrna (early 2nd century a.d.) 
cites Eratosthenes’ PhiUmicus for the same incident of Plato and the Delians 
with the same basic point: 

Ami Plato told them that not tor want of a double altar did the god pronounce this 
to the Delians, hut rathei to accuse and reproach the Greeks lor their neglect of 
mathematics and thcii slighting of geometry/ 

Aware of Plutarch's source, we thus do well to receive with caution the details 
of his account. It is clear, lor instance, that die chmnological association of this 
story with the uprising in Thebes derives entirely from Plutarch’s adaptation of 
the older version. 

Similar precautions arc necessary in dealing with another passage from Plu¬ 
tarch, providing a sequel to the Delian story by portraying how Plato reacted to 
the methods of solution worked out by the geometers. I his passage is inserted 
into Plutarch’s description ol Archimedes’ mechanical achievements, critical for 
the defense of Syracuse against the Roman siege of 215-212 n c 

This admired and much talked of instrumental art was first set in motion hy those 
around Eudoxus and Aivhylas, who embroidered geometry with subtlety, hut who 
supported hy means of perceptible and instrumental models problems not permitting 
rigorous gcomclric demonstration; for instance, in connection with the problem 
about the two means in proportion, an element necessary' for many of the geometric 
propositions, both of them resorted to instrumental constructions by fitting out 
certain means-drawing apparatuses from cwvcd lines and segments. But Plato took 
offense and contended with them that they were destroying ami corrupting the good 
of geometry, so that it was slipping away from incorporeal and intelligible things 
toward perceptible ones and lieyoml this was using bodies requiring much wear¬ 
isome matm lad me In this way mechanics fell out. separated from geometry, and 


t • ..«*»( »• Mi t*l Ul 


• IIKI4K i llinlLlllN 


for a long time it was overlooked by philosophy and became one of the military' 


Plutarch thus alleges a schism between the mechanical arts and the theoretical 
disciplines, bridged only by Archimedes, presumably for putting into practice 
his insights into pure geometry. The roots of this division Plutarch quite rea¬ 
sonably assigns to the idealist recommendations of the Platonic philosophy. 
Indeed, in Plato’s Republic , Socrates criticizes geometers for their insensitivity 
to the true nature of their inquiry: 

They speak in a very ludicrous and restricted manner. For they make talk about 
squaring and applying and adding and all, in a way as if they were engaged in 
practice and making all their words for the sake of practice. But surely the entire 
discipline is studied for the sake of knowledge.. ami knowledge of what always 
exists, but not of anything that can ever be made or destroyed (527 a-1>). They 
use the visible forms and make words about them, but their thinking is not about 
these things, but about what these things resemble, when they make their words 
about the square itself and the diagonal itself, but not about the one which they 
draw, and the others in this way; these very things which they fashion and draw, 
and which have shadows and images of their own. they can use them in turn as 
images in their search to see those things which no one could see other than hy 
the intellect (510 d-c).* 

It is this Platonic sentiment, this conviction of the purity of the objects of titic 
geometric study, which Plutarch captures in his anecdote on the cube duplication. 
Nevertheless, we can perceive that here too he draws front Kratosthencs as an 
intermediary source. For the Archimcdcs-cominentator Kutoeius of Askalon (6th 
century A.l) ) assigns to Eratosthenes a long account of the history of this problem, 
including the following passage as sequel to Plato’s meeting with the Delians: 

And when they |sc. the geometers in the Academy) applied themselves diligently 
and sought how to lake the two means of two given lines. Archytus of Tareiitum 
is said to have discovered it by means of half*cylinders, while litidoxus |did so| 
by means of the so-called curved lines. But it happened that all of them effected 
(his apodiclically, hui couldn’t construct it physically or pul it into service, save 
to some small degree Mcnaechimis. and that with difficulty." 

Here, as in Plutarch’s version, the geometers busy themselves under the watch 
of Plato, Archytas coming up with one solution, Eudoxus with another. Yet 
there is a profound discrepancy: Kratosthencs deems their solutions to be too 
abstract and impracticable, while Plutarch portrays them as overly mechanical 
and too little abstract. This* kind of discrepancy is one form of evidence adduced 
by some scholars to discredit Kutoeius* assignment of this writing to Kratos¬ 
thencs. 1,1 We will examine this argument later. For now, it is enough to note 
that the passages can be reconciled in a straightforward way. Two different 
writings by Kratosthencs arc at issue here: the document on the cube duplication 
and the Piatomcus\ it is the latter which Plutarch exploits as his source. The 
Platonicus was presumably a fictional essay, so that its strongly antimechanical 
position here could have resulted from Kratosthencs* adaptation of the other. 


.Sitting History Irani legend 5 


more plainly historical account, where the early solutions had no pronounced 
mechanical element. 

On the relative value of the mechanical and the theoretical elements in 
geometry, it is the attitude of the geometers themselves we would most like to 
know. Hie difficulties in teaming this from the passages just cited arc evident, 
since the glosses hy Plutarch. Kratosthencs. and the 4th-ccntury Platonists must 
first lie separated out before the underlying mathematical views can he discerned. 
Although Plato assigned a prominent role to mathematical studies within his 
general program of philosophy, that in itself does not qualify him or his disciples 
to speak for the technical researchers on the issues which concerned them. Indeed, 
passages like those cited from the Republic suggest that Plato and other philos¬ 
ophers might set themselves apart as critics of the technical field. But if we 
might try to infer from such critiques that the geometers insisted on a this-worldly 
orientation in their studies, we have the variant testimony of Kratosthenes tin 
the account of the cuhe duplication) that the early studies were cast in an abstract 
manner. Tlie mathematical lexis preserved hy Kutneius on the methods used by 
Archytas and Mcnacchmtis hear Kratosthencs out on this point. 11 Yet Kratos¬ 
thenes* own solution is accompanied hy the detailed account of an actual physical 
device built on its design, and this overtly mechanical element is found in the 
methods used hy Nicomcdes and Diodes and others around the same time (late 
3rd century n t ) ' ** Kviilently. views on the importance of mechanical approaches 
varied over the course of antiquity and might sometimes he a matter merely of 
personal preference. In this lack of unanimity on issues of metluKlological com¬ 
mitment. the ancient field of mathematics would conform to a pattern familiar 
in more recent periods in the history of mathematics." 

'These reflections thus reveal three major recommendations hearing on the 
present study of tlu* ancient problem-solving efforts First, we must attempt to 
establish the simple chronological sequence of the various solutions; only thus 
can one grasp for each period the complex of problems, methods, and aims 
which set the directions of geometric research at that time. On this basis one 
may hope to discern changes over the longer term in the character of the field 
of research. Typically, treatments of these materials have subscribed to a division 
hy s|>ccific problems or hy solving methods. Precedents can already be cited 
from antiquity in the compendium of solutions to the cube duplication presented 
hy Kutocius in his commentary on Archimedes* Sphere and Cylinder . in the 
sections on cuhe duplication and angle trisection in Books III and IV of Pappus* 
Collection , and inestimably also in the survey of circle quadrature in Sporu.s* 
Kvria , and in the surveys of sjHreial curves in the encyclopedias of Mcnclaus 
and Gcininus." Among modem discussions, one is most likely to consult the 
extended chapter on “Special Problems” in T, L. Heath s History of Creek 
Mathematics , the detailed articles on cube duplication, angle division, and tteusis 
constructions hy R. Hokcr in the Pauly Wissowa ReubEncyctopitdu\ the long 
paper on ancient studies of curved lines and surfaces hy P. 'Tannery, and the 
massive compendium on the history and properties of special plane curves by 
G. l.oria." Without questioning the great value of these accounts, as well as 




f» Ain.lent I litthiM 111 tumiictnc I'imIiJciiis 

many others too numerous to cite here, lcverthclcss believe that their principle 
of organization implicitly commits ihci- to the view that the ancients likewise 
conceived of the study of these particula curves or these particular problems as 
distinct subliclds. That might hold for an encyclopedist like Geminus, or even 
for commentators like Pappus or Kutocius, but not, it seems to me. for the 
original investigators like Archimedes and Apollonius and their contemporaries. 
By contrast, if we adopt a chronological scheme, we can suspend commitment 
on the ancient conception of the status of these problems and the various solving 
techniques, so that this conception will be allowed to emerge. 

The “simple** chronological questions are in fact often quite difficult to 
determine, as we shall find. The restriction to the three “classical” problems is 
of course artificial itself and will not be adhered to strictly. Its obvious advantage 
lies in the ample documentary coverage of these problems, touching, in some 
form, on each of the major periods of the ancient geometry. This raises the 
question of why the three problems continued to attract attention, even after 
many different solving methods had been worked out. An important key to the 
answer is found in the changes affecting the general field of research from one 
period to the next. As new techniques were introduced, their applicability to the 
older problems would naturally come up for consideration: for instance, a special 
curve generated in the context of a method for the angle trisection might be 
recognized to have properties relevant for an alternative solution of the cube 
duplication as well. 1 '’ It is thus clear that one cannot adequately comprehend the 
approaches adopted in the eases of the individual problems without full regard 
to the more general environment of research, litis ptcscnls not a difficulty, 
however, but an opportunity. For the documentation extant representing the more 
advanced researches at various times is often quite sparse. One may thus view 
the special problems as a cross section, suitable lor filling out one’s portrait of 
the wider field. This investigation will inrcupy the next six chapters of the present 
study (Chapters 2-7), moving from the pre-Euclidean |>criod ilimugli the gen¬ 
erations of Euclid and Archimedes to the time of Apollonius. 

A second recommendation is to devote separate treatment to several important 
metamathcmatical issues: how the ancients divided the geometric field according 
to the types of problems and solving methods; what they viewed the special role 
of problems to be, especially in relation to that of theorems, and how they 
associated these with the important methods of analysis and synthesis; what 
conditions they imposed on the techniques admissible for the solution of prob¬ 
lems, and whether they judged that satisfactory solutions for the three special 
problems had actually been found. Historical accounts of these materials almost 
invariably assume at the outset that the ancients had at a very early time distin¬ 
guished compass-and-straightedge constructions from the others ami were con¬ 
tinually urged on in the quixotic search for constructions of this type for the 
three special problems. 17 As we now know, through results obtained in the 19th 
century, the ancients were in a position neither to produce such solutions nor 
even to show, save through persistent failure, that they could not be produced. 
For the latter is essentially an algebraic question not amenable to investigation 


Silting History from legend 7 


under the geometric methods of the ancients. '* I believe, however, that we cannot 
assume that the ancient views on these metamathcmatical issues were just the * 
ones familiar within the modem field, or that even if certain geometers at certain 
times held views somewhat like those now familiar, all other geometers at all 
other times necessarily held the same views. Surely our goal must be to discover 
what the ancient views were by means of a consideration of the relevant historical 
evidence. This is the project undertaken in Chapter 8. However, as our look at 
Plutarch’s passages on the cube duplication suggests, the search for answers will 
be far from straightforward. Most of the passages we may consult come from 
later authors whose primary interest is in the explanation of methodological 
observations made by philosophers, most notably Plato and Aristotle. For in¬ 
stance, the nco-Platnnist element pervades the important Euclid commentary by 
Proclus (5th century a d ), while much of our evidence must come in the form 
of interpretations of specific passages from Aristotle by Alexander of Aphrod- 
isias, Themistius, Philoponus, Simplicius, and others. Recall that our passages 
from Plutarch derive from his discussions of Plato, and the same is true for his 
source, Eratosthenes. Even with writers who one might suppose arc closer to 
the actual mathematical tradition, such as Pappus, one can readily detect the 
influence of certain philosophical commitments on their mathematical 
pronouncements. 

This aspect of the commentators helps us to understand an emphasis commonly 
but, I believe, mistakenly made in discussions about the ancient geometry. One 
seems typically to assume that metamathcmatical concerns were the effective 
motivating force underlying the efforts of geometers, for instance, that the hall¬ 
mark of their tradition was their organization of geometric findings into light 
structures of deductive reasoning, as if their primary ambition was the production 
of treatises like the Elements of Euclid and the Conics of Apollonius. Hut this 
surely cannot Ik* correct. 'Hie writing of textbooks is the etui of mathematical 
research only in the sense that death is the end of life: it is the last term in a 
sequence, but not (be purposive element which explains why one engages in the 
activity of progressing through the steps in (his sequence. 1 * As far as the ancient 
geometers arc concerned, the goals underlying the activity of research arc in 
most cases a matter for surmise. In general, I will find most convincing the 
“internalist” position that technical research is directed toward the solution of 
problems arising from previous and current research efforts. In effect, the for¬ 
mulation of problems keeps a field open by guiding new research. To be sure, 
the goal to rigorizc (he proofs of known results can stimulate one form of research; 
the most notable example of this among the ancients may be found in the 
mathematical insights of Eudoxus on which were built the formal theories of 
proportion and limits.™ But with the exception of Eudoxus, the context of the 
interesting mathematical insights by (lie ancient geometers lies not in such formal 
concerns, but rather in the investigation of problems. In this respect, I believe 
that an intuition founded on the most recent period of mathematical history can 
be misleading, for mctamathematical inquiries have crystallized into a recognized 
subficld, the study of foundations, commanding expertise of an entirely math- 







H Ancient I ludiHuii i»l tiromctllC lloliJcuis 

ematical kind. But ihis did not happen in antiquity. If, for instance, the deductive 
project embodied in Euclid's Elements entails subtle questions of axiomatics, 
the ancients seem to have rest content with Euclid's formulation and with the 
Aristotelian theory to which it loosely relates, for both remain the focus of later 
discussions of foundational issues. 21 

The third recommendation is to take up the textual issues bearing on our use 
of the writings of authors from later antiquity. Their editorial efforts have pre¬ 
served for us most of what now survives of the ancient geometric tradition. In 
some instances, they produced the extant editions of the treatises of the great 
geometers like Euclid, Archimedes, and Apollonius; in others, they compiled 
miscellanies of geometric results drawn from works now lost. It is thus extremely 
important for us to gain a sense of how these writers handled texts: Where did 
they obtain their texts? What changes did they make in them? What were their 
purposes in selecting and presenting these texts? In this regard, the texts on the 
three special problems, in particular, that of the cube duplication, provide an 
ideal instrument, since they survive in multiple versions from several writers. 
It is remarkable that so little along these lines has been attempted up until now. 
Even those prominent scholars, J. L. Heiberg and P. Tannery, for whom phil¬ 
ological issues were a particular concern, did not much cxploic those questions 
which the comparisons of these texts might elucidate. It is as if the later writers 
were of no intrinsic interest to them, and that once one of several related texts 
could be accepted as the “best/* the others might be disregarded . 2: In a sequel 
following the present volume an attempt will he made to display the interrelations 
among the surviving texts of these solutions. One can well anticipate that these 
writers are not highly original in their mathematical contributions. But the degree 
to which they depend on sources, not only for mathematical techniques and 
results, but even for the very wording of the texts they present, I believe, will 
be surprising even for those who have made a study of the ancient geometry. 
This dependence applies as well to Pappus as to the other commentators, despite 
the diversity and advanced level of the materials in his Collection. Further, the 
pattern of textual methods will be seen to carry over among Arabic writers of 
the 9th and 10th centuries aj>. The discussion of several texts on the angle 
trisection and cuhc duplication from Arabic authors, in comparison with selected 
items from the earlier Greek tradition, will occupy a part of the sequel and its 
Appendices. There I will attempt to show that many of these Arabic texts were 
produced as translations, or at least close paraphrases, of the Greek, even in 
certain cases where one might have supposed they were original Arabic writings 
or reworkings of Arabic prototypes. Thus, the derivative character both of this 
pari of the Arabic mathematical tradition as well as of the Greek editorial tradition 
from the Graeco-Roman and Byzantine periods becomes clear and suggests that 
one might hope to exploit them in new ways for gaining insight into the earlier 
creative phases of Greek geometry, litis last is of course a project extending 
well beyond the scope of the present study. 

In this survey of the ancient problem-solving efforts, attention to matters of 
textual evidence will predominate in Chapter K and the sequel study. Elsewhere, 


Silling Histoiy limit l cgtmi 9 


the essential mathematical aspects of the different technical contributions will 
provide the principal instrument for revealing their historical relations. A diffi¬ 
culty here is that the ancient writers (especially those in the later editorial tra¬ 
dition) preferred the synthetic mode of exposition in (heir formal treatments of 
geometry. That is. one derives a claimed result (c.g.. theorem or construction) 
as the conclusion of a deductive chain of reasoning which starts from the given 
terms of the theorem and otherwise utilizes only axioms and other first principles 
or previously proven theorems and constructions. To mathematicians of the 17th 
century, indeed, even to some of the ancients, this format was notorious for 
obscuring the essential line of thought, particularly in the cases of more com¬ 
plicated results. 21 But when the discovery of solutions for geometric problems 
was at issue, the ancients exploited an alternative method: that of analysis . In 
the case of a problem ol construction, for instance, one here posits the desired 
figure as having already been effected and then deduces properties of this figure 
until an clement of it emerges which is known from prior results to be construc¬ 
tive. The formal synthesis would then begin from these constructive terms and 
proceed through a deductive sequence in approximately the reverse order to that 
of the analysis until the desired construction lias been produced. Since the analytic 
method can be applied fruitfully even where the solution of the pmhlcm is not 
yet known, if is especially well suited lor the purposes of geometric research. 
If the ancients sometimes followed this analytic method in their presentations 
of known solutions as well, this would doubtless owe to the appropriateness of 
exposing learners to the use of this method in preparation for their own research 
efforts later. It also has advantages for exegesis, seeing that the analysis of a 
problem exposes in a natural and well motivated way the rationale behind each 
step of the construction. By contrast, when only the synthesis is given, these 
steps often appear arbitrary and without clear motivation until much later in the 
proof. Thus, in my accounts I will prefer analytic presentations to synthetic ones. 
In cases where only the synthesis is extant, I will fashion the analysis corre¬ 
sponding to it. Given the close parallelism between these two parts of the treat¬ 
ment of problems, one can recover such analyses with virtually absolute confidence. 
In doing this, moreover, one can sometimes discern reasons for otherwise puz.- 
z.ling features in the extant syntheses. Nevertheless, the reader should be aware 
that this is an interpretive element in the account which follows. My intent is 
not to paraphrase or otherwise reproduce what is already available in the extant 
primary literature, hut rather to bring forward the essential, usually simple, 
geometric idea underlying each result mid thus to provide an appropriate intro¬ 
duction for further investigation of that literature. 

A lew terms admitting a variety of meanings in common usage are so prom¬ 
inent throughout this discussion of ancient geometry that I do not dare employ 
them in any sense other than their formal mathematical one. This applies to 
"analysis” and "synthesis,” for instance, which will here signify only those 
special geometric methods mentioned in the preceding paragraph. Similarly, the 
term "problem” will refer only to the geometric form of a problem, that is, to 
the type of proposition which seeks the construction of a figure in answer to 




10 Ancient Tiadiiion ol Ucomcinc Problems 


Silting History from Legend 11 


certain specified conditions or the detennination of certain properties of a spec¬ 
ified figure. It will never be a mere synonym of “difficulty," for in fact a 
geometric problem need not be difficult at all. Its essence lies in its being a 
"project" (from protmllein, “to cast forward”), that is, a demand to do some¬ 
thing not yet done or to discover something not yet known. Another such term 
is “prove" along with its cognate noun "proof and the synonyms "demon¬ 
strate" and "demonstration.” respectively. Here these refer only to those de¬ 
ductive forms of reasoning employed in mathematics and logic. I will forego 
their use in metamathematical and historical contexts, where more flexible terms 
like "show.” "reveal," "make evident." and so on, will be used in their stead. 

The serious student of ancient mathematics knows well that Ihe speculative 
component is necessarily very high. Virtually all the interesting questions we 
might care to investigate are beyond the range of definite answers, and the same 
applies even in the instances of many quite trivial matters. It is certain, for 
instance, that Apollonius wrote the Conics', but the question as to how much of 
the extant text is by Apollonius himself, rather than by his editor Eutocius, has 
not yet received the attention it demands. In view of this, one's assertions about 
these materials must always be received within a range of plausibilities and 
probabilities. A phrase like "one thus secs...” signifies a claim of a rather high 
degree of likelihood, whereas one like "it may well be... ” signifies a low one. 

An oddity of our language is that ostensible intcnsiliers like "surely,” "cer¬ 
tainly," and "doubtless" actually indicate quite the opposite of certitude, by 
signalling a situation within which doubts can he raised; their lorce is largely 
suasive, but will at least communicate to the reader the nature of my spiritual 
commitments to my claims. If l sometimes lapse into incautious use of the 
indicative mood, the reader should be forewarned to supply the needed ,! 

qualifications. i 

In the matter of my interpretive method, I subscribe, however imperfectly, . 

to three simple ideals: to consider all the available elements pertinent to a given j 

question; to advocate that view, among all the viable contenders, which best 
conforms with the general character of the ancient geometric tradition ami with 
the most straightforward conception of the mathematical field in relation to other 
disciplines; to prefer that view among viable alternatives which most enhances , 

the interest value of the extant documents. Applying these principles can often j 

be a delicate matter. In the case of the first, the.mere survival of a document 
need not compel us to believe what it says. If two commentators happen to | 

disagree on a certain point, we must of course try to determine which (if either) i 

is correct and to explain why there is a discrepancy. By the same token, if the i 

statement of only one commentator is known, that need not be true. Here the ; 

second principle assists in ascertaining whether a testimony might he suspect. i 

If, for instance. Pappus asserts something about the geometric field which flatly 
disagrees with what we find in the technical literature, surely we would not 
hypothesize a lost corpus of writings merely to make his view correct, but rather, 
we would admit his error and try to figure out why he made it. 24 One can hope 
eventually to gain a sense of the intrinsic credibility of the various authorities 


in their various contexts. One learns quickly, for instance, that Archimedes 
commits no technical errors and that any assumption to the contrary on our part 
would be merely presumptuous. 

The second principle serves to introduce an assumption of economy in inter¬ 
pretations. It is of course possible that ancient mathematical thought differed 
from that of other periods in certain striking and fundamental respects. But I 
believe we must be forced to such conclusions, rather than concur in them 
carelessly or even as a matter of principle. It seems to me, for instance, that the 
technical activity of problem solving in mathematics and other fields is never 
particularly sensitive to external factors, like the opinions of one or another 
school of philosophers. To be sure, the line of division may be difficult to draw 
sharply in some cases, and what start as external considerations might sometimes, 
through a very subtle process, become part of the internal conditions of research 
activity. But 1 do not imagine the ancient geometers as constantly looking over 
their shoulder. It is surely absurd to suppose that Platonist notions of the per¬ 
fection of circles induced mathematicians to restrict their solving techniques to 
compasses; or even that Zeno’s paradoxes of motion instilled in Eudoxus a fear 
of the infinite. The notion that ancient mathematics was somehow a vast exercise 
in dialectical philosophy must miss a very important point: that geometry is 
rooted in an essentially practical enterprise of problem solving. 25 

As for Ihe third principle. I believe that scholars have a responsibility to keep 
alive the interest in the documents they examine. It seems to me quite wrong to 
assume that a certain document is a forgery, unless the reasons arc compelling, 
for that will merely guarantee that no one takes it seriously thereafter. 2 '’ Again, 
it seems to me wrong to assume (hat a certain mathematical writer is incompetent, 
unless we are driven to (hat view on clear grounds. On the other hand, it is 
equally important to avoid romantic notions about the expertise of our authorities 
in the face of clearly contradictory indications. The stature of Pappus as an 
outstanding mathematical intellect, lor instance, has prevented the intelligent use 
of his writing as a source for the earlier geometric tradition. 2 ’ Again, if one 
begins to doubt that Aristotle himself produced the extraordinary mathematical 
reasonings in the Meteorohgira, this work might eventually be returned to the 
mainstream of thinking about ancient mathematical science. 2 " 

It is perhaps in the interest of scholarly precision to approach general theses 
about the nature of a discipline with scepticism. But most of the major treatments 
of the ancient geometry, such as that of Heath, have taken that wise counsel loo 
far. They set before us what purports to be a factual narrative of the ancient 
field, but which is hardly other than a melange of details without strong signs 
of interrelation, or even of internal consistency. The most exciting aspect of this 
material, in my view, is that a closer examination reveals the general lines of a 
coherent development, and that Ihe extant record, despite its incomplete state, 
actually makes sense as the remnant of an extraordinary movement in thought 
whose basic outline is discernible. If the reader remains unconvinced of the 
details of the argument in the following study, yet nevertheless comes to ap¬ 
preciate the unsatisfactory state of current scholarship and to perceive the pos- 










12 Ancient Tradition of Geometric Problems 

sibility of achieving a coherent view of this movement, then my effort shall have 
attained its principal objective. 

NOTES TO CHAPTER l 

' One may consult the Oxford Closured Dictionary (2ml cil.. Oxford. IWI) lor a 
brief account and bibliography of the life and work ol Plutarch. 

» Morolin 579 a-d; cf. the text, translation and annotations on this and related 
passages by P. II. dc Lacy and B. liinaison. Plutarch's Mnralia (Cambridge. Mass./ 
London: Loch Classical Library). VII, 1959. pp. 396-399. 

■ The ritual background to mathematics is a prominent concern in several articles by 
A Seidenbere: cf. his ‘’Ritual Origin or Geometry,** Archive for History of hum 
Sciences 1963. pp. 488-527 and his "Origin of Mathematics." Archive for History of 
Exact Sciences. 1976. 18. pp. 301-342. Some of these ideas have been taken up and 
extended by B. L. van dcr Wacrdcn. "Prc-llabylonian Mathematics. Archive for History 
of Exact Sciences. 1980. 23. pp. 1-46. The thesis of n philosophic background to pre- 
Euclidean mathematics, especially in the form of Bertie inlluenccs on the Pythagoreans 
has been championed by A. S/abd; cf. Anfdnge der gnechmhrn Maiheniaiik. Budape.t 
and Munich/Vienna. 1969 (translated as The ReRinnings of Greek Mathematics. Dor¬ 
drecht 1978). I have presented a detailed critique ol S/.aho’s views in ° n the Parly 
History of Axiomulics " in Pisa Conference (IV7HI Proceedings, ed. J Hmt.kka el al.. 
Dordrecht. 1981. I. pp. 145-186. 

4 The restriction to given lines A. 2A is ol course inessential; one may m general 
posit lines A. /». «. any given ratio and obtain A*: X* - A : «. as the ancient treatments 
of the problem .ire well aware. 

4 Cf Eratosthenes, as cited hy liutocius in Archimedis Opera, ed. J. L. I'rite'JL 
III. p. 88; cf.. also. Proclus. hi Euefidem. ed. G. I riedlcin. Leipzig: Icuhncr. 1873. p. 
213. Hippocrates’ work is discussed in Chapter 2. 

‘ Thenn. Eipasiua rerun i mathcmaiiciiriim. ed. K. Hiller. Leipzig: Teuhuer. 1878. 
p. 2. 

’ Plutarch, hives: M,ocellus, xiv. 5-6. An alternative version of this anecdote, men¬ 
tioning the geometers Archytas. Eudoxus, and Mcnacchmus. appeals m Plutarch s ty 
Plato said that god always geomclriz.es * (Moralia. 718 c-0. one of several d.seuss.ons 
on topics related to Plato’s Tinmens. Certain theological overtones present in tins version 
of the Delian story must be due to Plutarch's elaborations of his source narrative, just as 
in the Marcellos passage the concern to link this story with historical (tends in geometry 
and mechanics must also he his. 

* One may compare the lest, translation and comments by P. Shnrey in Plato: I lie 
Republic (Cambridge. Mass./London: Ijicb Classical Library). 1935. II. pp I70L 1121. 

• Archimedis Opera, ed. J. L. Heiberg. III. p. 90. 

Notahly. von Wilamowitz: cf. Chapter 2. 

" Archimedis Opera. III. pp. 78-88: these arc discussed in Chapter 3. 

“ Cf. Chapter 6. 

" Consider the following bold sounding claim on the "practical purpose of geometric 
investigations”: 


Sitting Histoiy hom legend 13 


The graphical solution of problem* of construction forms a major goal of geometry. Even 
where this goal seems in a certain degree to have been forgotten through lire advancing of 
theoretical investigation, its guiding influence on this investigation is quite easily perceived. 

The slatcntenl is made hy the Italian geometer. I*. Enriques, in the article concluding the 
anthology of contributions by hint und collnhoiatois oil advanced questions pertaining to 
elementary geometry. Question! rif»uardtuiti la yeometria elentenntte (2nd ed., 1907; I 
have followed the second (ierman edition. 1923. II. p. 327). 

By contrast, another mathematician has this to say about the ancient sludics on the 
conic sections; 

I believe dial one did not in general effect practically this construct ion (of die cubic toot| 
via conic sections, hut that indict it was a ilicorctical means—as it can also he for us— a 
means fnt the extension ol understanding \l\ikautfnis\ 

(From H. O. Zeuthen, “Die geomctrischc Constmetion als ‘Existenzbeweis* in der an* 
liken Geometric,” Mathemattsche Annalen, 18%, 47, pp. 222f; this essay will be dis* 
cussed further in Chapter K.) One may expect that the same division of opinion on the 
relative roles of practical constructions and theoretical investigations would still mark 
modem discussions on the nature of mathematics. 

II Kutocius, in Archimedis Opera. Ill, pp. 54 106. Pappus. C alien ion. ed. Ilultsch. 

1, pp. 54-68. 270 284. On Mcnclniis. Get limits. and Sportis. see I Heath. History t»f 
Greek Mathematics . I. pp. 226. 22%. H, pp. 222*226. 260f. I will propose further views 
m the sequel. 

M Heath. History of Greek Mathematics, Oxford, 1921, I. Ch. 7. Boker, “Winkel 
und Krcistcilung,” Pauly Wissowa. Ser. II, 9 M 1961, col. 127 150; *'Wiirfclvcrdop 
pclung." thtd., col. 1193 1223: "Nrusis." Pauly Wiswwa. Suppl. IX. 1962. col. 415-* 
461. Neusc.s are lamitiar m the form of sliding marked ruler constructions; see Chapters 

2, 5. and 6. Tannery. “Pour Phisfoirc des lignes ct surfaces coitrbcs dans I'antiquitl,” 
Hall. .Vci. Math. Astr.. 1883. 18. 278. 291 and 1884, 19. 19*30, I0M »2. Loriii, Hhene 
Kurven: Throne und Grschichte. 2nd ed.. l.cip/.ig/Bcrlin. 1910 (1st Italian ed. 1902; 
2nd Italian ed. 19091 

** Cf. the discussion of Nicontedes* conchoid in Chapter 6. 

11 Cf. Heath’s treatment in History. I. pp. 218-220. A distinguished exception to this 
tendency is the long study hy A. D. Steele. “Ueber die Kollc von Zirkcl und Lineal in 
der gfieehischcn Muthemalik.” QueUen taulStudten, 1936. 3, pp. 287-369. Steele blends 
a careful examination of the ancient sources with a good grasp of the modem algebraic 
view of these issues; he argues persuasively that the restriction to compass and straightedge 
was only rarely the explicit condition of the ancient study of geometric problems (cf. 
Chapter K). 

'* For a discussion of the rcducibility of third order problems together with a detailed 
survey of methods of cube duplication and angle trisection, one may consult the article 
by A. Conti in F\ Enriques, Frauen der Hlementar^eometrie. II (cited in note 13 above); 
for a discussion of the transcendcntality of u. also with historical background, one may 
consult B, Cain in the same book. In the anthology Monographs on Topics of Modern 
Mathematics, ed. J. W. A. Young. 1911 (Dover, repr., 1955) one may find an article 
on “Constructions wiih ruler and compass” hy L. E. Dickson, and another on the 
“History and transcendence of by D. E. Smith. 



14 Ancient liadiiion ul Cicomcliic Piohlcim 

•• The pun is possible in Greek, where the word Itlos and derivatives carry the various 
senses of "end,” “end result.” and "goal”; cf. Aristotle. Physics. II. 2. 194 a . 

10 On Eudoxus, see Chapter 3. 

» The ancient field of the philosophy of mathematics is richer than my statement 
suggests. But that study seems primarily to have engaged philosophers, interested in 
accommodating mathematical questions within their own favored philosophical outlooks, 
like Platonism, Epicureanism, or Scepticism, lire participation of mathematicians within 
this field appears to have been slight, and its influence on their researches slighter still. 
This ancient philosophical field is the subject of recent papers and a monograph in progress 
by I. Mueller. 

« \i/ c have already alluded to an instance of this altitude, in the treatment of the two 
versions of the Delian story surviving from Eratosthenes. 

»• gee, for instance. Carpus’ remarks on the nature of geometric problems, cited in 
Chapter 8 Dissatisfaction with the ancient formal style led several 17th-century geometers 
to reconsider questions of method and notation; sec. in particular. Descartes. Gtomeinc. 
The method of analysis thus served to stimulate the development of new algebraic ap¬ 
proaches in geometry; cf. M. S. Mahoney. The Mathematical Career oj Pierre dettm*. 
Princeton University Press, 1973. Ch. II. I. III. V. 

** This example is not hypothetical, cf. Chapter 8. 

» Sec the view expressed by Enriques, cited in note 13 above. 

“• litis is clearly the effect which von Wilainowiu* rejection of the Eratosthenes 
document has had. discouraging use of its potentially significant testimony to the early 
history of cube duplication; cf. note HI above and Chapter 2. 

>» pappus' repute both as a mathematician and as a spokesman on the nature of 
mathematics is an issue ... Chapter 8 and will be further examined ... the sequel volume 

*• See Chapter 4. 


CHAPTER 2 


Beginnings 

and 

Early 

Efforts 


When did geometers first initiate the quest lor solutions of geometric problems? 
One seems to find some hints toward a form of answer from certain discussions 
by Proclus, who, as leader of the Academy in Athens in the 5lh century a n , 
included guidance in (he study of elementary geometry as pan of his (cachings 
in the Platonic philosophy. Ilis commentary on Euclid's Rook I is a rich source 
not only of ancient views on the philosophical aspects relating to mathematics, 
but also on details of its historical development. From Proclus wc receive several 
items on prc-Kiicliilcan efforts derived from the history of geometry compiled 
by Aristotle's disciple Eudcnnis of Rhodes toward the close of the 4th century 
U.C Two of these relate to problems of construction: 1. 12 (how to draw the 
perpendicular to a given line through a given point not on that line) and I, 23 
(how to draw an angle equal to a given angle at a given point on a given line). 
Reportedly, these problems were first solved by Ocnopidcs of Chios, a geometer 
and astronomer, elsewhere identified as an associate of Hippocrates of Chios 
and hence datable to after the middle of the 5th century d.c.' 

This witness to the work of Ocnopides has led scholars to wonder how it 
could he that such elementary efforts emerged so late. Assuming to the contrary 
that (hey must have been known much earlier, they infer that it was the specific 
form of the solutions, as found in Euclid's treatment, which Ocnopidcs origi¬ 
nated, so that lie would become the one responsible for die formal project of 
effecting geometric constructions within the restriction of employing compass 
and straightedge alone, as is characteristic of Euclid's method in Rook L Pre¬ 
sumably, then, Ocnopidcs had access to a diverse range of ways for solving 



16 Ancient Tradition of Geometric Problems 


beginning and Early lit Ini is |7 


these and other problems, and set about the task of regularizing and classifying 
problems according to the means of construction adopted. 2 

If such a bald statement of this thesis seems unsympathetic, a more subtle 
representation would merely camouflage its intrinsic intplausibility. To be sure, 
much older precedents for the sort of geometric construction assigned to Ocn- 
opides can be detected. Bor instance. Babylonian tablets from the mid-2ml mil¬ 
lennium B.c. display figures of circles and squares drawn with the assistance of 
instruments, while the Egyptians' use of cords in their practice of geometric 
mensuration earned for them the name "ropc-strctchcrs” among the later Greeks. 
But the type of inquiry supposedly engaged in by Oenopidcs is ol an entirely 
different order of formal sophistication: to effect constructions on the consciously 
formal restriction to a specified set of means. It is no obvious matter to explain 
how and why such a distinctly formal move in geometry should appear so early. 
Attempts to account for it within the context of an alleged formalization of the 
general geometric field through linkage with prc-Socratic philosophical devcl 
opmenis. as with the Sth ccntury Elcntics and Pythagoreans, arc fraught with 
difficulties of their own. 4 

In the case of Ocnopides, such doubts are magnified upon closer consideration 
of the details communicated by Proclus: specifically, that Oenopides introduced 
his construction of (lie perpendicular for its utility in astronomy, ami that for the 
phrase "at right angles" l>e employed the term "gnomon-wise” (tow gnomonti), 
that is. with reference to the "gnomon" or pointer of a sundial. It thus seems 
evident that Kudenius, Proclus' source, has gleaned his information Irom a 
treatment of astronomical constructions known under the name of Oenopides.' 
But then it hardly follows that these were the earliest constructions of these 
problems, even under the special manner of construction they employ, but only 
that these were the earliest such instances of them known to Budemus. More 
importantly. Ocnopides would surely not have been occupied in a consciously 
formal geometric effort, for the astronomical context indicates that he was show¬ 
ing how to arrange the construction of astronomical devices, like sundials, via 
appropriate instruments of construction. l'l\c latter surely included compasses 
and rulers, but we have no reason to suppose that he rejected the use of others. 
Such devices as set squares, forms «»f compasses and sectors, angle-measuring 
devices, plummets, and sliding marked rulers were all in the repertory of tech¬ 
niques available to geometric practitioners in antiquity, and attested in the math¬ 
ematical literature from the decades before and alter Euclid.' 1 

It is imperative, I maintain, to raise these doubts about the formal nature ol 
the work of Oenopidcs and his contemporaries. While wr might consider it 
obvious and natural to classify constructions according to the means employed 
and to assign privileged status to those demanding only compass and straightedge 
for their execution, our intuitions in such matters arc thoroughly conditioned 
through knowledge and adoption of the objectives of the formal geometric tra¬ 
dition, advanced primarily through the works of Euclid and Apollonius. But in 
fact this formal restriction on the treatment of problems in itself betokens at¬ 
tainment of a sophisticated ilieorctical level. Thus, a historical inquiry into the 


study of problems must recognize the sophistication implicit in this move and 
ask about the time, manner, and motive of its introduction. Most treatments of 
ancient geometry, and specifically those dealing with the history of problem 
solving, assume this formal motive at the outset.’ This assumption, I believe, 
presents serious obstacles against gaining insight into the development of the 
ancient studies as they actually h;tp|icncd. Ixt us then set aside that assumption. 
Through the examination of the evidence of the early work, specifically on tire 
two problems of cube duplication and circle quadrature, we may hope to discover 
the roots of the ancient field of problem solving. 

THE DUPLICATION OF THE CUBE 

We possess accounts of the early stages of the study of the cube duplication in 
two reports derived from Eratosthenes of Cyrene, a prominent man of science 
and letters from (lie latter part of the 3rd century B.c One of these is the fragment 
from a scene in his dialogue, the PUuonicus, and is preserved by Theon of 
Smyrna and Plutarch in writings from the 2nd century a ij." The other lakes the 
form of a letter addressed by Eratosthenes to his royal patron Ptolemy HI Eucr- 
getes in association with a transcript of the description of an actual model of his 
device, the "mesolabc" ("mean-laker"), for the mechanical solution of this 
problem and the text of ait epigram dedicating it in the temple of the Ptolemies. 
Hie latter account is preserved as one of eleven texts on this problem compiled 
by Enlocius of Askalon. the fith-century-A i> commentator on Archimedes' woik. u 
The value of such a document for examining the early history of these efforts 
would seem evident; hut it has not been exploited by historians, owing to their 
general acceptance of (lie case challenging its authenticity."’ It is thus important 
first to consider whether that ease is to Ire sustained. 

lintocius* version runs to live full pages in the standard edition by J. L. 
Heiberg. We may divide it into live sections: (i) A historical introduction cites 
a scene from a tragedy about Minos by an unnamed dramatist and a legend 
telling of Plato's involvement witlr the oracle of Delos as precedents for an 
interest in the problem of doubling tire cube. Specific note is made of efforts by 
Hippocrates of Chios and by three associates of Plato; Archytas, Eudoxus, and 
Menacchmits. (ii) Eratosthenes' mechanism for finding means is cited and praised 
for its greater practicality in comparison with the earlier methods. Several con¬ 
texts of potential application arc enumerated, among them ship building and 
military engineering, by way of illustrating lire utility of the device, (iii) A full 
account of tire geometric theory of Hie device is given, (iv) This is followed by 
a description of certain physical details pertaining to the actual constniction of 
a working model, (v) lire text closes with a description of tire votive monument 
raised by Eratosthenes to commemorate its invention: this consisted of a bronze 
exemplar of the mesolabc set atop a pillar with an explanatory inscription 
engraved below. A full transcript of the latter is given, including (a) a brief 
account of the geometric theory of the device, parallel to that in (iii). and (b) 
an epigram of eighteen verses singing tire praises of its virtues in contrast to the 






18 Ancient Tradition of Geometric Problems 

same prior efforts named in (i). and finally dedicating it to Ptolemy and his j 

son." „ I 

Since the penetrating examination of this document by U. von Wilamowitz- j 

Moellendorff in 1894. the authenticity of the closing epigram and of the de¬ 
scriptive text just preceding it |(v-a) and (v-b)l have been granted." The epigram 
is too expertly crafted, too fine a specimen of the difficult form of Hellenistic 
elegiac couplets, he argued, to be so easily dismissed as a forgery the way critics 
before him had contended. The descriptive text on the model of the device would 
raise no suspicions (save for an occasional trivial interpolation), while the event 
of an inventor or artisan's dedicating a work sample at a shrine is not without 
precedents in this period. But von Wilamowitz had no such approval to give to 
the accompanying letter. First, he argued, this presents an extended account of 
the device (iii) which would surely be superfluous, given the presence of the 
dedicatory model itself. Second, on points of detail it conflicts with the ''gen¬ 
uine" part of the text (v). lie alleged, as well as with the fragment from the 
Platonicus available to us through Thcon and Plutarch; on the whole, however, 
it appears to assert little beyond what one might have read in the inscription or 
inferred from that. Third, this text was not known to Pappus, who wrote about 
two centuries before F.utocius, for his own version of Eratosthenes' method 
differs on points of detail in the construction and proof from the one given in 
the letter, although it presents a highly condensed version of materials parallel 
to Eutocius* items (iv) and (iii) or (v-a). Finally, and most damaging ol all, 
the letter is banal: it is written without any sense of style and without any 
recognition of the occasion of its communication. One can hardly conceive, 
argued von Wilamowitz, that a man of such literary talents as Eratosthenes, 
writing to his patron, should frame his account in such a pedestrian manner. 

Thus, he concluded, the letter must he taken as a late forgery, still unknown at 
the time of Pappus; its author presumably transcribed from the votive monument 
the epigram of Eratosthenes and the partially mutilated description of the device 
and then, misled by the epigram's devotional invocation of Ptolemy as deity of 
the shrine, mistook the offering as a personal gill and so produced the letter as 
a companion explanatory document. But, he added, the ineptness ol the forger’s 
effort is betrayed on many points, most strikingly in its omission of second 
person forms of address almost until the very end. On the other hand, the 
composition has preserved for us the genuine epigram, lor which we may be 
grateful. 

Shall one accept this view of the provenance of the letter? To be sure, its 
style is concise, perhaps a bit abrupt; but it is not an illiterate production. One 
has yet to display such outright anachronisms which would mark it as impossible 
for a writer of the 3rd century n c. Moreover, the historical and technical in¬ 
formation it presents is entirely compatible with what we learn from our other 
sources on the early studies. Indeed, the absence of anachronisms relating to the 
mathematical content is impressive and would suggest at the very least a degree 
of thoughtfulness and skill on the part of the alleged forger. A comparison with 
the "genuine” parts of the document shows that the “forger" has not limited 


Beginnings and Early lit f uns 19 

himself to merely what has appeared in the epigram, but goes beyond it on 
several points in ways which indicate a real familiarity with this material, rather 
than mere fabrication. For instance, he observes that of the earlier methods only 
that of Menaechmus admitted any practical implementation “to a certain extent 
and that with difficulty” {tlyscherdsY. by contrast, the epigram refers only to the 
constructions by Archytas as “unwieldy" ( dysmiehana ), where in context the 
charge applies indifferently to the methods of both Menaechmus and Eudoxus 
as well. 14 

Relative to the argument concerning Pappus, no one has ever presumed that 
Pappus passed on all the sources at his disposal. Even if he did not possess a 
copy of the source in the form used by Eutocius. that hardly implies the nonex¬ 
istence of that source at Pappus’ time. After all. Eutocius himself presumably 
passed over the version in Pappus (if he had access to that) in favor of the text 
he actually reproduces. Surely, Pappus was capable of the same kind of editorial 
selectivity. As it happens, his text on Eratosthenes is rather more concise than 
that given by Eutocius, but on the whole not superior to it. Von Wilamowitz 
makes much of certain discrepancies (e.g., the shape of the sliding plates as 
triangular according to Pappus, but rectangular according to Eutocius)’' which 
have no bearing on the feasibility of the design and its realization. But he is 
j silent on certain insights of Eutocius omitted by Pappus: e.g.. that the "meso- 

j labe" can be used for finding not just two. but any number of mean proportionals 

by the insertion of additional plates. 1 '’ The notion that Eutocius’ report could 
result merely by transcribing and adapting what could be seen on the monument 
is simply not true, although it might well hold for the version in Pappus. In 
particular, Eutocius is far more detailed in his recommendations on the physical 
construction of the device. It is difficult enough to accept that the inscription 
should remain more or less intact as laic as Pappus’ time, early in the 4th century 
a I).": hut that such a memorial of pagan worship could survive the volatile 
spirit of the filter part of dial century and the next is truly incredible.Von 
Wilamowitz’ argument, that most of the information in the letter would be 
superfluous in the presence of the monument itself, is not compelling. One may 
consider the example of the ornate astrolabe invented and built by Synesius, 
disciple of Hypatia, statesman, philosopher and bishop; the specimen was for¬ 
warded to his noble associate Paconius accompanied by a letter describing in 
detail the conception and physical execution of the device. Many of these details 
would be evident from the device, including the two epigrams engraved on it, 
which Synesius quotes in full: "let it be set down for such as may read it later, 
since for you it is enough that it lies on the tablet.""’* One thus secs in Synesius’ 
dedication a close parallel to Eratosthenes’ letter on the "mesolahc." 

The strongest indication of the letter’s authenticity, however, would appear 
to be precisely what von Wilamowitz took to be its most questionable feature: 
its banality. For surely a forger would have let his imagination run wider than 
the narrow compass of factual materials given here. By contrast, those facts do 
amount to the kind of information which technical accounts accompanying other 
mathematical writings of this period contain. ‘Hie prefaces to the treatises by 



20 Ancient Tradition of Geometric Problems 


Beginnings and Parly Efforts 21 


Archimedes, for instance, rarely go beyond stating the principal theorems in the 
work and making brief observations on the methods used. ,v One would suppose 
that his correspondents Dosillteus and Eratosthenes were receiving these com¬ 
munications by virtue of their official positions at Alexandria; but even this 
inference is not directly affirmed by anything actually said in the prefaces. 2 *' The ! 

Sand-Reckoner provides a good parallel to Eratosthenes’ letter. It is devoted \ 

primarily to the cx|H>sition of the geometry and number theory needed for a j 

specific computation dealing witli the dimensions of the cosmos. 21 Within it are i 

details of the design of a sighting device by Archimedes himself and a report , 

of particulars, both geometric and observational, relating to its actual use. ITie 
discussion elaborates a theme introduced through a general literary reference ! 

(i.e., the inadequacy of the account of the infinite by certain unnamed philos¬ 
ophers). In all these respects, then, it follows a pattern like that of Eratosthenes’ 
letter. Indeed, both are framed as communications to royal patrons, yet the Sand- J 

Reckoner , despite its much greater length, is as sparing in its use of second- ! 

person forms as the letter is. 2 * The Gclon, addressed as “king” by Archimedes, . 

is believed to have served as regent in Syracuse lor some thirty years, although ; 

he never reigned as king in his own right. 2 ' Was he still a youth or already a j 

mature man when Archimedes wrote to him? Was the writing sent to him at j 

Syracuse, or elsewhere? And where was Archimedes himself at that lime: at ; 

home or abroad? Did he read the letter orally before an audience or was the j 

writing intended for Gclon’s personal study? Was Archimedes a kinsman of the I 

regent or his former tutor? The document provides not a single clue toward the 
answer to any of these questions of context. * 

In view of this, how can one fault Eutocius’ document for its failure to establish 
a setting? Von Wilamowit/.* critique thus falters through his own failure to reckon 
with the character of the appropriate literary genre. Without grounds for sus- j 

picion, then, we may treat Eutocius’ text no less seriously than any other, as a j 

historical source. Indeed, von Wilamowit/.’ plea for admitting the epigram, 
falling within a genre of which he is a recognized master, would naturally prepare 
us for admitting the whole document as genuine. The burden of proof rests upon 
those who would maintain the contrary. 

Under the view that the letter is authentic, its ostensible conflict with the ; 

alternative account from the Platonicus takes on entirely new significance. With j 

reference to the efforts of the geometers in Plato’s Academy, the letter says this: j 

(a) After some lime they say that certain Delians fell into the same difficulty as 

they set about to double one of the altars in accordance witli an oracle, and so • 

sending out word they asked the geometers with Plato in flic Academy to find for 

them what was sought. These applied themselves diligently and sought (how) to 

take two means of two given lines, |b| and Archylns of Taras is said to have 

discovered (a solution) by means of semicytindcrs. Eudoxus by means of the so- 

called curved lines. |c| But it happened that these all wrote in demonstrative fashion, 

none being able to manage it in a practical way or to put it into use, save to a 

certain small extent Menacchntus and that with difficulty/* 

It would appear that two or three different sources have been stitched together: 


(a) an account of the legend of the Delian oracle and the communication to the 
Academy; <b) a reference to solutions by Archytas and Eudoxus; and (c) an 
account somehow acquainted with writings derived from these geometers, in 
particular, from Mcnacchmus. Doubtless, communications between Archytas 
ami the Academy were consistently good, through visits by Plato and others to 
Italy, and through correspondence; but the ancient biographical traditions do not 
mention an actual residence by Archytas at the Academy. 21 Thus, line (b) reads 
uncomfortably as a simple continuation of (a). Further, both (a) and (b) are 
reported at some detachment from primary sources (“they say;” “it is said”). 
By contrast, (c) seems to speak from a certain familiarity with such sources. 
When Eutocius elsewhere presents Archytas’ method, Eudemus is named as 
source. 2 * One would suppose that Eudemus also treated the methods of Eudoxus 
and Mcnacchmus and that his versions were available to Eratosthenes. Thai 
heing so, the tentative tone of line (b) is puzzling: surely it would be obvious 
that Archytas used semicylinders and Eudoxus used curved lines. Mere then is 
a place where an inter)M>lator may have inserted a line, based on his reading of 
the allusions to these geometers in the epigram. 22 Without it, line (e) follows as 
a fully appropriate technical observation after the story in (a). 

The other account of this incident is transmitted by Thcon of Smyrna in the 
introductory section of bis compilation of mathematical materials pertinent to 
the study of Plato: 

For Eratosthenes says in liis writing the Plutonian that when the god pioiimmccd 
to the Delians in the matter of deliverance front a plague that they construct an 
altar double or the one that existed, much bewilderment fell upon (he builders who 
sought how one was to make a solid double of a solid. Then there airivcd men to 
inquire of this from Plato. But lie said to them (hat not for want of a double altar 
did the god prophesy this to the Delians, hut to accuse and reproach die Greeks 
for neglecting mathematics ami making little ol geometry.*’" 

This version ol the story is further attested in two passages from Plutarch, for 
whom the moral is that one needs dialectical skill to maneuver among flic 
ambiguities of oracles/’ Following von Wilamowit/, many have taken these 
discrepancies to militate against the authenticity of the version in Eutocius. One 
is thus to suppose that the fiirger of the letter diluted Eratosthenes’ account in 
the preparation of his own. But why should lie have sought to remove all those 
details which give the story vividness: the occasion of the plague and Plato’s 
personal intervention to interpret the (me meaning of the oracle? The story in 
the letter seems to have no point beyond explaining why a certain group of 
geometers came to concern themselves with cube duplication. 

The embellishments reported by Thcon and Plutarch suggest that their versions 
derived from the elaboration of the more |>cdcstrian account cited by Eutocius. 
Doubtless, Eratosthenes himself was responsible for these changes, as lie sought 
in his Platonicus to dramatize aspects of Plato’s view of mathematics. After all, 
motives of this very sort led Thenn and Plutarch to cite this passage in their own 
works. Elsewhere, Plutarch exploits the cube duplication to illustrate another 






"* Wrt-KB'H* rf .e#&y»» a «npK¥«»r tvTH^ r. -. •• 


22 Ancient Tradition of Geometric Problems 

aspect of Plato's philosophy: that geometry is not a matter of perceptible things, 
but only of the eternal and incorporeal. Thus, he has Plato criticize the mechanical 
solutions of Archytas, Eudoxus, and Menaechmus as spoiling the true good of 
geometry. 30 This alternative version echoes the second part of Eutocius' account, 
where the practicability of these geometers' solutions is brought up for comment. 
But there is a patent difference: according to Eutocius, Eratosthenes criticizes 
the earlier efforts for their being loo abstract; by contrast, Plutarch derives from 
him a criticism of their overly mechanical character. Of course, this discrepancy 
need only be one of emphasis. The surviving accounts of these methods arc in 
fact fully geometrical in the formal manner; but they do rely on certain mechanical 
conceptions, like the generation of curves by moving lines or by the intersection 
of solids of revolution.' 1 The student of Plato’s abstract philosophy might well 
view the latter with disapproval. On the other hand, Eratosthenes' own solution 
entailed the production of a mechanical device, and in his eyes its practical 
viability is a major asset; in this context, the practical limitations of the earlier 
designs could well be a (mint of note, inducing him to underscore their purely 
theoretical nature. Here again, the discrepancy is undoubtedly due to Eratosthenes 
himself. The intent in the letter is historical, but that in the Platonicus is dramatic 
and philosophical. He might thus take certain liberties in the latter, denied to 
him in the former. To set the scene for his dialectical point, he can elaborate 
the mechanical elements more explicitly than the original treatments actually 
did. Such fictionalizaiion would surely be acceptable in a work which made no 
pretension of being plainly historical. 

Accepted as a serious historical source by a man well placed for a knowledge 
of this period of Greek mathematical history, Eratosthenes' (ext as reported by 
Eutocius offers hope of insight into matters often obscured in modem discussions. 

-i First, although historians typically wish to leuve open the question of the historical 

r validity of the Delian oracle. M one now readily perceives the story to he a 
fabrication, originating from within the Platonic Academy around the middle of 
; the 4lh century n.c. By this lime the problem of cube duplication was already 

; familiar, for as Eratosthenes himself notes, important advances in its analysis 

I were made by Hippocrates of Chios, that is, almost a half-century before." It 

; would seem odd indeed for (he Delian oracle to be concerned with an old problem 

which then happened to be eluding the efforts of contemporary geometers. But 
on the other hand, as a dramatic way of affording recognition and motivation 
’ to those efforts, the story makes good sense, especially since geometers asso- 
ciated with the Academy were prominent in this activity. But one can only guess 
what the context and purpose of the story might have been at its first composition. 
As we have indicated, the specific morals attached to it by Eratosthenes (in the 
Plaionicus ) and by Plutarch seem to he their own additions, hut they are likely 
to have sensed accurately the ancient Academicians' view of these matters. Any 
student of the Republic can appreciate (he strength of Plato’s insistence on purity 
and abstractness of mathematical entities.' 4 The kinematic element in the con¬ 
structions for the cube duplication might thus provoke discussion of the relative 


Kt-giiiiiiiiKN ami luiily I-.M.mIn z.t 

status of geometry and mechanics, and indeed this very issue is important within 
Aristotle's theory of the order of the sciences.'' 

Furthermore, Eutocius' text of Eratosthenes is noteworthy for the sparseness 
of the historical information it transmits. One should have thought that Eratos¬ 
thenes would be concerned with producing as full an account as he could of the 
studies preceding his own, and his text docs leave that impression. Yet it is 
amazing that he turned up with si* little. He appears to have discovered nothing 
worth mentioning from the whole century or so separating him from the efforts 
of the Academicians. As for the even earlier efforts, he cites only the work of 
Hippocrates and a scene from an unnamed tragic poet.’'’ In the latter, when King 
Minos is told that the tomb he had ordered for Glaucus will have dimensions of 
one hundred feet on a side, the king answers that this would be too small, and 
so. "let it be double; without mistaking its fine form, swiftly double each member 
of the tomb." The new structure will of course be eight times larger than the 
original one in volume, not its double. Eratosthenes calls attention to the poet’s 
error. Rut actually there is none here: the poet intends that each dimension shall 
be doubled; he is not articulating the mathematicians’ problem of the cube 
duplication. Evidently, Eratosthenes has misconstrued the passage in his desire 
to find precedents for an interest in this problem. 

In effect. Eratosthenes has sought some motivation for Hippocrates' study of 
the cube duplication. But Hippocrates' effort by its very nature reveals that 
motive; for in Eratosthenes' account, this is what he does: 

It used to be sought by geometers how to double the given solid while maintaining 
its shape. .Alter they had all puzzled for a long time. Hippocrates of Chios was 
first to come up with the idea that if one could take two mean proportionals in 
continued proportion between two lines, of which the greater is double the smaller. 
then the cube will Ik* doubled. Thus he turned one puzzle into another one, no 
less of a pur/le." 

Hippocrates' insight is of course not restricted to lines assumed in a 2 : I ratio. 
If lor any two given lines. A and II, wc can insert the two mean proportionals, 
X and Y, then A : X = X : Y = Y : B. Thus, hy compounding the ratios, one 
has (A : X)' = (A : X) (X : Y) (Y : B). that is. A*: X' = A : II. Thus, X will 
be the side of a cube in the given ratio (B : A) to the given cube (A').'* 

In his closing line here. Eratosthenes scents to perceive little merit in Hip¬ 
pocrates move. But in fact, by this stroke Hippocrates has put the problem into 
a form permitting the application of a whole new range of geometric techniques, 
those of proportion theory. In general, this procedure of "reducing" (u/wgdge) 
one problem to another from whose solution its own would follow is a powerful 
technique of problem solving, a forerunner of the fruitful method of geometric 
"analysis." As Proclus tells us, Hippocrates was the first geometer known to 
have applied the method of reduction in the investigation of "puzzling dia¬ 
grams," i.e. difficult problems of construction. w Wc will sec later that Hip- 





24 Aitcicm IruililMMi ill Cicomciric Piohlems 


licguimnj£N and hiirly lilloiis 2.S 


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pocraics appears to have adopted a similar strategy in the investigation of the 
circle quadrature. 

Later geometers clearly recognized that such a reduction does not in itself 
amount to a solution of the problem posed. But would Ilipppocratcs already 
have made this distinction in the case of his treatment of the cube duplication? 
A passage from Aristotle suggests that he did. In de Aninui II. ( h. 2, Aristotle 
observes that one may define a term like “squaring" Uetraadnixmos) mil only 
by saying what it is (i.c., the equality of a square with a rectangle), but also by 
stating its cause (i.c., the finding of a mean proportional). Now, the latter 
refers to the reduction of the given problem of areas, such as one may read in 
Euclid’s VI, 17. and is the precise plane analogue to Hippocrates’ problem of 
solids. But the construction of this planar problem, whether in the form of a 
squaring (as in Elements II. 14) or in that of rinding a mean (as in VI, 13), was 
certainly already available to Hippocrates. A long fragment of his study of the 
quadrature of crescent-shaped figures survives, and the techniques of proportions 
are applied throughout to situations far more demanding than the squaring of 
rectangles. 4 ' Since the distinction between the reduction and the solution was 
clear in the instance of the planar case, the same must surely have been appre¬ 
ciated in the solid case as well. We may suppose, then, that Hippocrates and 
his followers continued their research into the cube duplication by seeking the 
construction of the two mean proportionals, where the latter cflort achieved its 
first clear successes through the discoveries of Archytas, Eudoxus, and 
Menacchinus. 

These considerations help clarify the way in which the cube duplication was 
first articulated as a problem. Within the study of the measures of plane figures, 
affected via the techniques of proportions, it would be perfectly natural for a 
geometer like Hippocrates to consider the case of two mean proportionals anil 
so perceive its relation to the problem of volumes, by analogy with the case of 
the single mean. In this way. the new problem emerges through tlte natural 
development of geometric research. Now, it sometimes happens that research 
moves in new directions in response to external stimuli: practical problems arising 
from commerce, government, anil engineering are a manifest basis ol much in 
the most ancient mathematical traditions: ritual might sometimes play a role: or 
philosophy might instil a sensitivity to formal questions. 42 1 he story ol the ITclian 
oracle has thus often been used as an example of such external motivation. But 
one can now see that the ancient evidence on the origins of the cube duplication 
affirms quite a different view: that the legend arose within Plato s Academy 
around the middle of the 4th century ti e . long after the problem had attained 
notoriety through the work of Hippocrates; and that Hippocrates himscll artic¬ 
ulated this problem as a natural extension of tlte geometric field of his time. In 
the absence of direct testimony to earlier work in the 5th century, one has no 
grounds for supposing a development of any other sort. 


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THE QUADRATURE OF THE CIRCLE 

In connection with the problem of constructing in a given angle a parallelogram 
equal to a given rectilinear figure (Elements I, 45), Proclus remarks: 

Having taken their lead from this problem, I believe, the ancients also sought the 
quadrature of the circle Is* if a parallelogram is found equal to any rectilinear 
ligiiic. it is worthy ol investigation wlicihcr one can prove that rectilinear figures 
are equal to figures Imuiid try circular arcs.” 

Although the words "I believe" reveal this view to he merely a surmise on 
Proclus’ part, rather than a conclusion bused on more or less explicit testimony, 
it is far from unreasonable. Indeed, there appears to be a certain sense of the 
problem of circle quadrature in remnants of the ancient Egyptian and Mesopo¬ 
tamian traditions as far hack as the middle 2nd millennium B.c and earlier. 
Specifically, the scribe of the Rhind Papyrus works out the area of a given circle 
lirst by subtracting 1/9 of its diameter and then squaring the result. 44 Thus, in 
effect, the circle is equated with the square whose side equals 8/9 the diameter 
of the circle. This rule is illustrated by a figure in which a square is divided into 
nine equal squares, where the diagonals of the four small squares at the comers 
are drawn in. In this way one secs at once that the circle might be approximated 
as 7/9 the enclosing square. One might conjecture that the scribe went on to 
subdivide each of the squares into nine squares and then tried to estimate how 
many of the 81 resulting squares combined to approximate the circle. But another 
approach might have been followed. Knowing that problems of areas often 
require computing squares and their roots, the scribe might have sought a value 
near 7/9 which resulted from a squaring. Since 7/9 = 63/81, where the latter 
is just short of 64/81 » (8/9) 2 , a quasi-dcductivc route might be possible for 
deriving the rule applied by the scribe of the papyrus. 

If such was indeed the basis of the ntlc, the scribe would also have a basis 
for recognizing the measurement of the circle as a geometric problem. The same 
would not be true within a tradition relying entirely upon empirical derivations 
of its geometric rules. It is hardly possible, for instance, that the Egyptians could 
have known the I : 3 ratio of the volumes of the cone and cylinder having equal 
altitude and base other than by observing that the fluid contents of a conical 
vessel can be emptied exactly three times into a cylindrical one. or by some 
comparable physical measure. 4 ’ But inaccuracies of manufacture, spillage, and 
other such factors would render this procedure approximate. There would thus 
be no linn demarcation between rules which arc exact (as is that for the ratio 
of volumes) and those which arc approximate (e.g.. the circle measurement). 
Certainly, within the early traditions the aim was to obtain practical procedures. 
The project of somehow producing a construction of the square equal to the 
circle seems remote from this objective, even if it may be perceived as entailed 





26 Ancient Tuuiition ol ^.icomcluc I'icibleins 


■ i 


within the results given. Its explicit formulation as a problem must then be sought 
elsewhere. 

We may gather that Greek mathematicians had hit upon this formulation of 
the problem before the close of the 5th century b.c. For in his comedy The Birds 
(staged in 414 b.c.) Aristophanes presents this lampoon of the astronomer Melon 
arriving as self-appointed civic surveyor of Cloud-cuckoo-land: 

If I lay out this curved ruler from above and insert a compass—do you see?— 

.. .by laying out I shall measure with a straight ndcr. so that the circle Irccoines 
square for you “ 

Now. there is no report of Mcton’s inquiring into the circle quadrature, and the 
passage here actually views him as somehow dividing the circle into quadrants. 4 ' 
But the force of the jest would seem to require that the audience perceive it to 
refer to a real problem under investigation by geometers at that time. This 
conforms with reports we have of certain efforts by Hippocrates, Antiphon, and 
Anaxagoras. 

Relative to Hippocrates, Aristotle tells that he proposed a false proof of 
the quadrature of the circle “by means of segments" or "hy means of lun- 
ules." 4 * The latter refers to Hippocrates’ masterful construction and quad¬ 
rature of crescent-shaped figures hounded by arcs of circles; we turn later to 
the question of the bearing this effort had on the circle quadrature. According 
to the account preserved by Simplicius, the 6ih-century Aristotelian com¬ 
mentator, drawing explicitly from a report by Kudcmus. Hippocrates prefaced 
his treatment with demonstrations of certain basic theorems on circles: 

He made his start and set down first some things useful for those |constnictions|. 
that segments of circles have to each other the same tniin as tltcir bases in power; 
and this he proved from having proved that the diameters liavc the same ratio in 
power as the circles— *** 

This passage notes explicitly that I lippocralcs proved these theorems. Moreover, 
it goes tin to supply further details of the derivation of the segment theorem in 
good agreement with a version given much later by Pappus."* We may thus Ire 
confident that toward the close of the 5th century, I lippocralcs already appreciated 
the essential concents of the deductive fonn entailed by the proofs of tlteorcms 
such as these. Now, the lines just cited include tlte statement of a theorem 
identical to one in Euclid's Elements (XII, 2): "Circles are to each other as the 
squares on their diameters." Simplicius' lenninology of "powers” (dymnneis) 
instead of “squares" {tctra^diut) reassures us that here he has followed the 
archaic Eudcittcan usage in his text, rather than improvise through his own 
knowledge of Euclid. 51 But wc encounter difficulty in supposing that I lippocralcs 
advanced the proof of this theorem on circles in its Euclidean fonn. For the 
limiting method on which that depends is due to Eudoxus, as we learn on the 
good authority of Archimedes. 51 Then what sort of proof could I lippocralcs have 
provided a half-century or more before Eudoxus? 

Aristotle's remarks on the false efforts at circle quadrature link I lippocralcs 



with two sophists from early in the 4th century. Bryson and Antiphon." What 
Simplicius reports of the latter may help us here. Antiphon proposed to construct 
a rectilinear figure equal to a given circle by a procedure involving the successive 
doubling of the number of sides or an initial inscribed polygon. For instance, if 
one starts with the inscribed square, erects the perpendicular bisector to each 
side, marks the point where each meets the circle, and connects each of these 
points to the adjacent vertices of the square, one will have erected an isosceles 
triangle on each side of the square, and their combination with it will produce 
the inscribed regular octagon. (See Figure I.) One may next bisect the sides of 
this octagon, ultimately resulting in the inscribed 16-gon; and so on. 

Continuing in this way. at the point where the area was exhausted, he said he will 
have inscribed a certain polygon in the circle in this manner, such dial its sides 
conform to the ate ol die circle on account of tlicir smallness. Since we aic able 
to produce die square equal to any polygon... because the polygon Iras horn 
produced confonning to the circle, wc shall also have produced a square equal to 
the circle." 

Antiphon's step yielding an arc confonning (ephamunein) to tlte sides of the 
inscribed polygon would appear to be drawn from notions familiar through the 
ancient atomists. l ; or them, the microscopic world of atomic reality might be 
qualitatively different from the macroscopic world as perceived by out senses. 
.Specifically, the circle appears to meet its tangent at a single point only, and 
we insist on this as part of the abstract conception of this configuration; but in 
their physical manifestation they will meet along a small line segment. This very 
question was discussed by Democritus and some of the sophists, notably Pro¬ 
tagoras; and one of Simplicius’ sources on Antiphon, the 3rd-century-A.L>. com¬ 
mentator Alexander of Aplintdisias. observes that this principle-- the circle’s 
meeting the tangent line at tine point only—was the one whose abolition lay at 


i 



the root of Antiphon's fallacious quadrature/ 5 We thus perceive a firm dialectical 
context for Antiphon's argument. 

The invalidity of his circle quadrature is evident, for it relies upon the actual 
"exhaustion" (dapanan) of the whole area lying between the circle and the 
polygons, while the indicated procedure of inscribing polygons will not do this 
if applied only a finite number of times.Hence, for Aristotle and ihc com¬ 
mentators, its refutation is not a matter lor the geometer to he concerned about. 
By contrast, modem scholars have tended to assign to Antiphon credit for the 
introduction of (lie important notion of polygonal approximation, essential for 
the ancient quadratures of curvilinear figures. 5 ’ But is this a plausible position 
to maintain? Given that Antiphon and Hippocrates were near contemporaries, 
the latter without doubt being rather the older, shall we suppose that the sophist, 
in the course of framing a muddled geometric argument to support his dialectical 
position, lent essential insights to the geometer? Or is it not more likely that he 
was drawing from and modifying procedures already familiar among geometers? 
The latter certainly typifies the later philosophical tradition, as represented by 
Plato, Aristotle, the Stoics, and the Sceptics, for instance, and their stance toward 
the technical sciences of their times. By contrast, it seems unclear why a geometer 
like Hippocrates should turn to the sophists for technical assistance in a theorem 
of this sort. The fact that Antiphon's use of this technical procedure lapses into 
patent fallacy, while Hippocrates requires Ihc same procedure for Ihc proof of 
a theorem implemented with full logical precision in his quadratures of lunutcs, 
merely confirms one's doubts about Antiphon’s originality in this technical context. 

If we assign to Hippocrates knowledge of the polygonal procedure preserved 
in our text of Antiphon's quadrature, then we obtain a good basis for recon¬ 
structing a proof he could have used for tire theorem on the ratios of circles. 
For one notes a subtle difference between Antiplioo’s method and the one used 
in the Euclidean proof of this theorem (XII, 2). Euclid obtains each polygon in 
the sequence by bisecting each of the arcs cut off by flic vertices of the polygon 
preceding it. Antiphon, on the other hand, builds each new polygon from its 
precursor by adding the isosceles triangles onto each of its sides. Now. if one 
considers two circles, each enclosing similar polygons constructed in (his way, 
since the initial polygons and the added triangles taken pairwise arc similar, 
these figures will be in the same ratio, namely that of the squares of the diameters 
of the circles. Hence, the sum of all these figures, extended indefinitely, will 
be in the same ratio. Since these sums constitute the circles themselves, it follows 
that they too have the ratio of the squares of their diameters, as Hippocrates 
asserts. 5 * 

From the strict logical viewpoint, of course, (his argument fails, for it applies 
to the infinite case a theorem on the ratios of sums which can be established for 
sums of only a finite number of terms, if even that, given the technique of 
proportions available to Hippocrates. 5 " But this approach is so strongly founded 
on clear notions of the continuity of magnitude that the difficulty might not 
necessarily have been appreciated at an early stage of these studies. We must 
assign to Eudoxus the first full awareness of this difficulty and the discovery of 


the method of limits by which it could he removed. It is noteworthy that the 
technique of sums we have just proposed is a standby within Archimedes' dem¬ 
onstrations; that Archimedes provides a proof, but only for the finite case (Con¬ 
oids and Spheroids, Prop. I); nevertheless, that he applies its infinite case in all 
of the planar and solid measurements presented in his Method.*' 

One cannot say for certain whether such studies of the circle quadrature can 
lie set much before the time of I lippocralcs. Writing in (lie 2nd century a.i>., 
Plutarch mentions that the 5lh-century natural philosopher Anaxagoras "drew 
Ihc quadrature of the circle" during a spell in prison/' This would place an 
awareness of the theoretical character of flic problem of circle quadrature to not 
much later than the middle of the 5th century tic It is an especially intriguing 
attribution, in that Anaxagoras insisted on the continuity of matter as the basis 
of the nature of things and seemed to perceive some of the mathematical im¬ 
plications of the principle of indefinite divisibility, Ihc very principle later violated 
in Antiphon's circle quadrature. To lie sure, Proclus reports that Anaxagoras 
"touched on many things relating to geometry. ”** But the ancient tradition 
otherwise assigns to him no specific interest in mathematics, and Plutarch’s 
testimony is quite casual, inserted merely to illustrate the point that the thinking 
man finds happiness even under trying circumstances. The subsequent notoriety 
of the circle quadrature might all loo easily color the testimony transmitted by 
later authorities. Thus, we must forswear using Anaxagoras’ alleged contribution 
as a sign of early interest in the problem. 

Front Aristotle we may gather that a construction of the circle quadrature was 
attempted on the basis of Hippocrates’ quadrature of lunules. For in his account 
of Ihc logical method of "reduction” (rr/mgdgr*) he remarks thus; 

Wc are die closer to knowing, lire fewer Ihc mean Icons in ihc syllogism. For 
instance, as Ihc circle together with the lunules is equal to a rectilinear figure, we 
arc dial much die closer to die squaring of die circle itself." 

Viewed in the light of Hippocrates’ reduction of the cube duplication to Ihc 
problem of tltc two mean proportionals, this passage seems to indicate that 
Hippocrates attempted a similar strategy for the circle quadrature: to cast it into 
an alternative form permitting tltc application of a wider range of the techniques 
then known. This is fully bor ne out in the extended discussion of circle quadrature 
presented by Simplicius commenting on a related passage from the Physics where 
Aristotle mentions the false quadratures.** In addition to sketching and criticizing 
Antiphon's argument on the circle quadrature, as wc have already seen, and 
noting several other efforts on this problem, Simplicius gives a detailed account 
of Hippocrates’ quadrature of lunules. Indeed, he provides two different versions, 
a simplified treatment known through Alexander's commentaries, followed by 
an elaborate series of constructions and proofs quoted verbatim from Eudcntus’ 
history. This passage from Eudcmus is our most important fragment from the 
pre-Euclidean geometry, providing invaluable insight into the content, methods, 
and terminology or the early tradition. Wc shall examine it in detail, after a brief 
consideration of the shorter version from Alexander. 





Figure 3 



(a) 


Figure 2 


in Alexander’s text, two cases of lunulcs are prcsenlcd*":(a) Starling from a 
right isosceles triangle, we draw on cacti of its sides a semicircle. (See Fig. 2a.) 
Since the square on the hypotenuse equals the sum of the squares on Il*c two 
legs, and since circles arc as the squares on their diameters, it follows that the 
semicircle on the hypotenuse equals the two semicircles on the legs. If we now 
remove the portion common to the large semicircle and the two smaller ones, 
we find that the isosceles triangle which remains is equal to the two lunulcs 
which arch over its legs. Thus, hy equating the lunulcs to a rectilinear figure, 
we have effected their quadrature.** (b) In the second case, we begin with the 
trapezium formed as a bisected regular hexagon and draw semicircles on each 
of its sides. Since the longest side of the trapezium is double each of its shorter 
sides, the large semicircle equals fmir of the smaller ones. As before, subtracting 
the common portion, we find that "the remaining lunulcs together with the 
(small) semicircle are equal to the trapezium.” (Sec Fig. 2b.) 

Recalling Aristotle's remark on the reduction of the quadrature of the circle 
to that of the lunules, cited above, we see (hat he evidently had this construction 
of Hippocrates in mind. Hut precisely what conclusion Hippocrates himself 
intended to draw from it has been debated among ancient and modem com¬ 




mentators alike.*'’ Alexander, for instance, supposed that Hippocrates wished to 
argue that since the lunulc in (a) has been squared, while the sum of the circle 
and the lunulc in (h) has also been squared, then the circle itself has liecn squared. 
If so, he would have committed the patent error of assuming that the quadrature 
of a single case of lunulc amounts to the quadrature of all cases. Simplicius, by 
contrast, found it incredible to conceive that a geometer of Hippocrates' out¬ 
standing caliber could Itave lapsed into such a blunder. Under this view, one 
would charge to others, tor instance, to Aristotle, the attempt to construe Hip¬ 
pocrates* results as a circle quadrature. Doubtless, this is the correct view, for 
as we saw in the parallel instance of the cube duplication, Hippocrates' reduction 
of the problem there was not likely to have been regarded as a solution of it. 
As we shall see later. Simplicius' text provides further insights into Hippocrates' 
view of the circle quadrature. 

The alternative account, which Simplicius quotes “word for word" {kata 
Icxin) from Kmlcmus. presents the full treatment of four cases of lunules. The 
first is essentially identical to Alexander's case (a), while the fourth hears com¬ 
parison, if somewhat more loosely, with his case (b). The other two cases arc 
new, constructing lunulcs which arc in the one instance greater and in the other 
less than the arc of a semicircle. Translations and technical summaries of this 
text are readily available, so that a brief summary will suffice here."" My principal 
aim now is to retrieve the line of thought underlying Hippocrates' constructions 
and thus to reveal his sense of the more general class of lunulcs. 

<i) In the lirst case, Hippocrates begins with the right isosceles triangle and 
circumscrilics the semicircle on its hypotenuse. (See Fig. 3.) This much conforms 
with case (a). Hut now, instead of drawing semicircles on the legs also, he draws 
on the hypotenuse the circular segment similar to those which appear on the legs 
of tlic triangle. In this way he obtains a single lunulc. the figure hounded by the 
two circular arcs, and shows (hat it equals the initial triangle. This follows, since 
the similar segments arc as the squares on the chords which define them, so that 
the larger segment equals the sum of the two smaller ones. If to the mixtilincar 
figure inside we add the Iwo small segments, we obtain the lunule; if we add to 
it. alternatively, the large segment, we obtain the triangle. Thus, the triangle 
and the lunulc are equal to each other. It is clear that this lunulc is die same as 
each of the two lunulcs constructed in case (a). But the present treatment reveals 
a greater technical range. For instance, its author operates freely with similar 



Library of Congress cataloging in Publication Data 

Knorr, Wilbur Richard, 1945 - 

The ancient tradition of geometric problems. 

Bibliography. 

Includes index. 

1. Geometry-History. 2. Geometry-Problems, Famous. 
I. Tide. 

QA4435.K58 1986 516.2*04 83-7159 

ISBN: 0-8176-3148-8 


GP-Kurztilelaufriatime der Deulsclicn Riblinlhck 
Knorr, Wilbur Richard; 

The ancient tradition of geometric problems/ 
Wilbur Richard Knorr. - Boston; Basel; 

Stuttgart: Birkhauscr, 1986. 

ISBN 3-7643-3148-8 


All rights reserved. No part of this publication may be 
reproduced, stored in a retrieval system, or transmitted, 
in any form or by any means, electronic, mechanical, photocopying, 
recording or otherwise, without prior permission of the copyright 
owner. 


€> Birkhguser Boston, 1986 
ISBN 0-8176-3148*8 
ISBN 3-7643-3148-8 


Table of Contents 


Preface vii 

1 Silting History front legend I 

2 Beginnings ami Parly liflorts 15 ^ 

(i) The Duplication of the Oihc 17 

(ii) The Quadrature of the Circle 25 

(iii) Problems ami Methods 39 

3 The Geometers in Plato’s Academy 49 

<9 Solutions ol the Colic Duplication 50 

tii) Geometric Methods in ilic Analysis of Problems 66 

(iiil FI toils toward the Quadrature of the Circle 76 

(iv) Geometry ami Philosophy in the 4th Century 86 

4 The Generation of Hnclid 101 

(i) A I jioiv Problem in ihc Aristotelian Corpus Iti2 
di| Hoi*Iiil\ Analytic Works KIR 

tiiil *n»e Analysis of Conic Pmlilcim: Some Kccoiislnictinns 120 
nvl An Angle Trisection via "Surfaced .ocus" I2H 
tv> I* lie I id \ Conn Unit ion to Ihc Study ot Problems 137 









B 0 

Figure 4 


segments, not just semicircles. The pertinent definition, given earlier in the text, 
stipulates them as subtending equal angles in their respective circles. As wc saw 
above, his proof that similar segments, like similar rectilinear figures, have the 
ratio of the squares on homologous sides signifies that major progress has already 
been made toward the Budoxean manner of measuring curvilinear figures. In all 
these respects, the treatment of case (i), despite its relative simplicity, prepares 
us nicely for the more intricate constructions which follow. 

(ii) Hippocrates now assumes the construction of a trapezium whose base has 
to each of its three remaining equal sides the ratio \/3 : I. (See Fig, 4.) He 
next draws the circumscribing circle and then draws on the base the circular 
segment similar to those which have thus been formed on the three other sides. 
The text establishes in full detail that the arc of the outer hounding circle is 
greater than a semicircle.** The actual proof of the quadrature of the lunulc is 
omitted from Budemus* text, so Simplicius supplies his own, remarking that 
Eudemus must have considered it obvious. But in fact the analogous proofs arc 
given in full for each of the following (wo eases, and textual considerations show 
these to be in the pre-Euclidean manner of Eudemus. 70 Thus, one may infer that 
Simplicius' (ext came to lose portions of the proof through scribal omissions. 71 
At any rate, since the one large segment equals the three small ones, the same 
process of addition as used in (i) shows that the lunulc equals the trapezium. 

From these two cases we may recognize the condition Hippocrates has in 
mind for the construction of squarahle crescents. If wc conceive of a rectilinear 
figure bounded by two polygonal arcs, the outer consisting of n sides each equal 
to the inner consisting of m sides each equal to s m% and if s* : s m 2 = m : 
n, and if furthermore the sides .v,„ .s m subtend equal angles in (he circles circum¬ 
scribed about their respective polygonal arcs, then the lunulc bounded by these 
circular arcs will equal the rectilinear figure bounded by ihe polygonal arcs. 
Although Hippocrates docs not formulate the construction in such general terms, 


t 


i 

i 

i 


s 


r 

♦ 


E 


H 



(b) Figure 5 

his treatment of (he third ease indicates that a conception of this sort did indeed 
guide bis thought. His examination of this next ease, like those which preceded 
it, is entirely in the synthetic mode, and so lends to obscure the manner of its 
discovery. I shall thus attempt to reconstruct on its basis an alternative version 
in the analytic mode in order to display its essential idea. 72 

I Ait an investigation of the lunulc of the type just staled be proposed, where 
the two hounding arcs correspond to polygonal arcs, the one of three equal sides, 
the other of two. Thus, s 2 : .v, = \/j : v'2. Let us conceive that the figure has 
been produced, consisting of the double arc EZH and the triple arc EKBII (see 
Fig. 5a); if wc pass a circle through each of these polygonal arcs, then wc must 
also have that the circular arcs EZ, Zll, EK, KB, BH all subtend the same angle 
at the center of their respective circles. We join El I and consider the angles ZEH 
and KEH. As the former intercepts the single arc ZH, while the latter intercepts 
the double arc KBH, it follows that angle KEH is double ZEH, so that EZ bisects 
angle KEH and its extension must pass through B, Furthermore, point Z lies on 
the perpendicular bisector of HK r \ while point E lies on the circle of radius EK 
and center K. Thus, given lengths in the ratio \/3 : \/2 . to accomplish the 
construction wc need only take the shorter length as KB, draw the circle having 
the same length as radius and center at K, draw the perpendicular bisector of 
KB, and (hen so place a line passing through B that the portion of it intercepted 
between the circle and the perpendicular bisector equals the larger of the given 
lengths (see Fig. 5b). In this way, both the placement and the length of EZ will 
be determined, and from it the rest of the figure can be completed. 

The latter sets out precisely the same form in which I lippocratcs accomplishes 





Ihe construction of case (iii) of the luiuilcs. In particular, one perceives through 
this analysis how natural it is to use the placement of EZ for effecting this* In 
Hippocrates* terms, “the line inclining (neuousa) toward B and intercepted 
between the line DG and the arc AEB shall have the given length.*’ 74 This is 
the earliest known instance of the use of the constructing technique called neusis 
by the Greeks. As we shall later see, it plays a prominent role in many of the 
problem-solving efforts by Archimedes and his followers a century and a half 
after Hippocrates. 7 ' 

Having so constructed the lunule, Hippocrates next proves that it is equal to 
the associated rectilinear figure EZHBK; for since 3 EK 2 = 2 EZ 2 , the three 
outer segments will equal the two inner ones. Since, furthermore, “the meniskos 
(lunule) consists of the three segments together with that part of the rectilinear 
figure apart from the two segments/* 76 the lunule and the rectilinear figure are 
equal* Hippocrates goes on to show that the outer bounding arc of this lunule 
is less than a semicircle. At this point Kudcmus injects an amazing remark: 


In ihis manner Hippocrates squared every lunule. since in fact iciper) |hc squarcdl 
the | lunule] of a semicircle and the one having the outer arc greater than a semicircle 
and the one lhaving the outer arc| less than a semicircle.. .. Hut lie squared a 
lunule together with a circle as follows . 71 

What arc we to make of “every lunule**? Docs it signify merely the three 
particular cases just examined? But in context a more general sense seems to be 
indicated, since one moves at once to the consideration of the case pertaining 
to the circle quadrature. Then has Eudcmus erred in supposing that squaring one 
lunule or even a few is tantamount to squaring them all? If thi*s is so, then did 
Hippocrates himself make the same error, thus bringing on himself Aristotle’s 
charges of having produced a fallacious argument on the circle quadrature? But 
it is hard to imagine how the one responsible for these carefully ordered proofs 
on the lunulcs could then have committed such an elementary blunder in rea¬ 
soning. Now, there is indeed a sense in which Hippocrates might claim to have 
established the quadratures of a whole class of lunulcs, namely those for which 
the m similar segments along one bounding arc equal the n .similar segments 
along the other arc. Although, of course, he has not produced the general con¬ 
struction of these figures, he can assume that their quadratures are obvious by 
virtue of the three cases given. If this is what Hippocrates intended by asserting 
that “every lunule** had been squared, wc would infer that Kudcmus, and doubt¬ 
less others before him, had misconstrued his claim as referring to every possible 
form of lunule. 

As for case (iii), so also for (iv), the synthetic mode adopted by Hippocrates 
conceals its essential line of thought. But it emerges if one views this case in 
relation to case (b) in Alexander’s version, by analogy with the relation of cases 
(i) and (a). If wc start with the hexagon inscribed in a circle, instead of using 
either its side or its diameter as hasc, let us intn>duce the chord connecting the 
two nonadjacent vertices II and I. (Sec Fig. 6.) Thus, III is the side of the 







inscribed equilateral triangle, while NT is the side of the hexagon, so that 111 : 
HT = \/3 : I. If wc draw over 111 a segment similar to each of those over HT, 
T|, it will equal three limes either of those segments. In the familiar maimer, it 
follows that the triangle HTI equals the lunule together with one of those smaller 
segments. Thus, squaring the lunule reduces to squaring that segment. Hippo¬ 
crates conceives of a set of segments similar to it and equal to it in sum; these 
are the segments on the sides of a hexagon inscribed in a circle whose diameter 
has to that of the initial circle the ratio I : Vfi. The segment over HT thus equals 
the sum of all six segments on the sides of the hexagon ABGDEZ. Adding to 
both the area of the hexagon and then combining with the earlier result, wc find 
that the triangle together with the hexagon is equal to the lunule plus the smaller 
circle. In Kudcmus* text the construction begins with the drawing of the two 
circles, so lhal the motivation behind the choice of I : as the ratio of their 

diameters remains unclear until well into the proof. Having established the 
equality just staled above, he concludes: 

Thus, if the rectilinear figures |i.c.. the triangle and the hexagon) mentioned can 
be squared, so also can the circle with the lunule/" 

This ends Ihe quotation from Kudcmns; for Simplicius observes in the very next 
sentence: 

One ought to place gicatci mist in Kudcmus (than in Alcxandcrl to know things 
pertaining to Hippocrates of Chios, for lie was nearer to his times, being a disciple 
of Aristotle. 

It is dear in Kudcmus* account that Hippocrates makes no presumption of having 
squared the circle itself, but only the circle in combination with the lunule. 'fins 






Figure 7 


(a) 

is so clear, in fact, that Simplicius questions the view that Hippocrates intended 
this here at all, and so suggests that when Aristotle speaks or his fallacious 
quadrature of the circle “via segments,” he must refer to an entirely different 
argument.™ Far more likely, however, is that Hippocrates presented this last 
case of lunulc as a successful reduction of the circle quadrature, without its 
being a resolution of that problem. He would thus leave for later research the 
investigation of this lunulc. If some followers rashly asserted that the circle had 
indeed been squared in this way, we have no grounds at all for charging this 
error to Hippocrates himself or even for suggesting that he wished slyly to leave 
this impression. 

Aside front Hippocrates' attitude toward the circle quadrature, we should 
consider what his view of constructions was. It will be helplul to consider lirst 
the more general class of lunulcs and certain modern contributions to their study. 
At each end of the given line segment 00' let there be drawn a system of equally 

spaced rays, OA, OB, OC, 0D.and O'A\ <I'B\ 0 'C . 0'IV.respectively. 

where in each ease the angle 0 separating consecutive rays is the same. (Sec 
Fig. 7a.) Then we may form a polygonal arc from 0 to 0' by selecting the 
intersections of appropriate rays from each system. For instance, if point a is 
the intersection of rays OA, 0 f B' and point b is that of OB, O'A', then the broken 
line O'abO will be a polygonal arc consisting of three equal lengths; for if we 
pass through them the circular arc joining 0 and O', each length will be a chord 
subtending angles equal to 0 relative to the points 0 and O'. Ihns. the circular 
segments having these chords as base will be similar to each other. (Sec Fig. 
7b.) 110 In analogous fashion, we can construct between 0. 0' another polygonal 
arc having as many equal elements as desired/ 1 As before, we obtain a set of 
circular segments similar to each other and to the segments associated with any 
other such polygonal arc constructed via the same initial sets of rays. Lei two 
such arcs be taken, the inner having n elements, the outer having m elements. 
Denoting by X a side of the inner arc (c.g., O'a) and by Y a side of the outer 



(b) Figure 7 


arc (e.g., O'c), and applying the identity of sines to the triangles 00'a and 00'c, 
we have X : Y = sin ##i0 : sin nil. The condition that the lunulc bounded by 
the two corresponding circular arcs be squaruble in Hippocrates* sense is that 
tiX 2 =mY* , so that sin mO : sin rid = Vw : Vw. If the general case is denoted 
by the pair («, /n), then the three eases examined by Hippocrates are (1, 2), (1, 
3), and (2, 3). These all happen to be constructiblc in the familiar ‘‘Euclidean** 
sense, that is, by means of compass and straightedge alone. That two other cases, 
(1,5} and (3, 5), arc of the same sort was noted by Wallenius (176b) and Clausen 
(1840)"*; for one can readily see that their defining relations reduce to a quadratic 
in cos 20. Furthermore, researchers have established that the lunulc (l, p) is not 
constructiblc in this manner when p is a prime not of the Gaussian type (that is, 
of form 2 7 l I) (Landau, 1903), nor is the lunulc («. p) so constructiblc for n 

“1,2 . p I (TschakalolY, 1929)/'Moreover, even for Gaussian prime 

p (other than 2, 3, and 5), the lunulc (1, p) is not constructihie (Wegner); 
furthermore, apart from the cases already known to be constructible, the lunulc 
Or, tn ) is not cnnsimctihlc when «. m arc both odd integers (Tschcholarbw. 
I935). M 

For our purposes now, it is important to realize that this modern conception 
of the problem of constructing the Itmules is anachronistic as far as Hippocrates' 
study is concerned. For we recall that his construction of lunulc (2, 3) depended 
on a neusis, that is, the suitable inclination of a line segment of given length. 
As it happens, an alternative mode for effecting the neusis he requires is possible 
via planar methods (i.e., compass and straightedge), for it may he expressed as 
a quadratic relation falling within the class of problems known as the ‘‘hyperbolic 
application of areas,” so that a solution can Ik worked out on the model of 
Euclid's Data, Prop. 59, based on Elements II, 6 and VI, 29/* While it is a 
reasonable conjecture (hat Hippocrates had access to these techniques, nothing 
in our text directly supports that claim. Certainly, his manner of introducing the 
neusis gives no hint of requiring us to effect this step by any means other than 
the obvious: the manipulation of the marked line EZB/ h Indeed, even later 




geometers like Archimedes Irecly admit neust \t in elicit constructions, where 
sometimes, as in Spiral Lines, Prop. 5, an alternative ••Euclidean” method is 
possible/ 7 Thus, I would maintain that Hippocrates* goal was to find any con¬ 
struction he could. If he proceeded from the general conception of the config¬ 
uration given above, he might perceive that the construction of any lunule (n, 
m) satisfying the condition X : Y = \/m : y/n reduces to a determination of 
the appropriate angle 0. Initially (for 0 = 0) the ratio X : Y equals m : «. By 
experience of actually constructing figures, he could realize, if not prove, that 
as 0 increases the ratio decreases, and hence that the specific configuration sought 
is possible/" In principle, his admission of neuscs as a constructing technique 
makes accessible to him a wider range of cases than just those of the ••Euclidean** 
type, including, for example, cases reducing to cubic relations.** Thus, the fact 
that he presents only the three sqnarablc eases indicates that neither he nor any 
other ancient geometer who tried his hand at this problem was able to come up 
with any cases besides these. In ibis instance, ihcn, not having recourse to the 
later algebraic methods in geometry constituted a real barrier against the discovery 
of solutions lo these problems. 

As noted above, Simplicius assigns greater weight to Eudenius* text than to 
the one transmitted by Alexander, believing that the former more closely rep¬ 
resents the work of Hippocrates himself. Now, from the purely textual point of 
view, he seems quite correct in maintaining this. In certain contexts, lor instance, 
Eudemus adheres to older terminology discontinued in the tradition after Euclid: 
he speaks of “powers** Ulynameis) instead of “squares** (tetragona), and uses 
expressions like “the line on which (eph' hei) AB** about as frequently as their 
short forms (c.g., “the line AB,’* or simply “the AB**), while only the latter 
arc found in the later formal tradition™ By contrast, Alexander's text is cast in 
full conformity with Euclidean usage. On the other hand, from the technical 
viewpoint, Alexander’s constructions of lunules (a) and (b) arc rather simpler 
than the analogous eases (i) and (iv) of Eudemus; and the technical refinement 
evident throughout the proofs id* Eudemus* four eases clearly surpasses the 
treatments in Alexander. It is of course conceivable (hat Alexander's text orig¬ 
inated from an effort to simplify an earlier version in the manner of Eudemus. 
But we recall that in his passage aw “reduction,” Aristotle alludes loaquadraiurc 
set in the form of Alexander's case (b), rather than Eudemus* case (iv). This 
affirms that Alexander's version too derives ultimately front a 4th-century source. 
Its simpler treatment of (he lunules well suits what one would expect of the early 
phases of such a study; and as we have seen, ii assists considerably in retrieving 
the motivations underlying the more elaborate constructions given by Eudemus. 

This raises ihe question of the provenance of Eudemus* text. Much depends 
on Eudemus* reliability and sophistication as a historian. While we possess far 
too little of his work lo make a secure judgment on this matter, it would seem 
that the purposes occasioning his account of Hippocrates would be admirably 
served through a fair transcript of his source materials, while Eudemus* motives 
for producing a radically modified account, indeed even his competence to do 


so in such a detailed technical context, would he entirely unclear. Can we then 
assume that Eudemus* text reflects with reasonable accuracy an actual writing 
of Hippocrates? There is of course no difficulty in supposing that Hippocrates* 
initial version of the study of lunules was in the simpler fonn indicated by 
Alexander, and that he later produced the more refined version recorded by 
Eudemus. On the other hand, it is equally plausible that the latter version grew 
out of the researches of followers inspired by his efforts to pursue this study. 
Doubtless, Eudemus was in the position to distinguish between a genuine writing 
of Hippocrates and a later writing in (he tradition instigated by him. But would 
he insist on making this distinction clear in his discussion of (he quadratures? 
We can hope that he would, yet must admit that it is possible he did not/ 1 
These considerations affect one’s use of the fragment as an index of the form 
and terminology of 4lh-ccnlury mathematical writing, for we cannot wilh cer¬ 
tainly locale ii closer lo the time of Hippocrates, toward Ihe beginning of that 
century, than to (lie lime of Eudemus, nearer its end. Nevertheless, ils reliability 
as a testimony of Hippiurraics* findings seems secure, lor we have Eudemus* 
explicit slalcniciil several times in the fragment that the four eases of lunules 
were all studied by him, and it is hard to suppose that he could have been 
mistaken in this regard. Furthermore, the proofs of the quadratures, even in the 
simpler version reported by Alexander, indicate advances in the geometric tra¬ 
dition around Hippocrates’ rime; access to a wide range of theorems and con¬ 
structions of rectilinear and circular figures, results of (he geometric theory of 
proportions, and so on, as well as appreciation of the basic concerns in the 
deductive ordering of proofs. Bui regardless of whether Hippocrates did indeed 
work oul the impressively formalized synthetic treatments given by Eudemus, 
rather than, say, a more loosely framed analytic treatment, his contribution to 
(he study of these geometric problems is unquestionable. 

PROBLEMS AND METHODS 

This survey of the earliesi efforts within the ancient tradition of problem solving 
has revealed (lie central importance ol the work of Hippocrates of Chios. In 
particular, his studies of (he cube duplication and of the quadrature of the circle 
and related figures laid the foundation for later researches. According to Proclus, 
he was also responsible for the first compilation of “Elements,” that is, of a 
systematized presentation of propositions and proofs/ 2 From the surviving frag¬ 
ments of his work, we may infer that this included substantial materials on the 
geometry of circles (cf. Euclid's Book III) as well as an exposition of the 
properties of plane figures dominated by the techniques of proportion theory (cf. 
Book VI), even if of course there were still lacking Ihe more rigorous methods 
of limits and proportions (comparable lo Books V and XII) due to Eudoxus. 

Although stories like the Delian oracle on the doubling of (lie cube are often 
cited as suggesting an external motivation for the early studies of such problems, 
our examination of Eratosthenes* account of them and our inquiry into Hippo- 





crates' work serve to discount this view. On the contrary, the interest in these 
problems can be seen as a fully natural outgrowth of researches within the 
geometric field. For instance, having cast the problem of squaring a given rec¬ 
tangle into the form of finding the mean proportional between its length and 
width, Hippocrates could be led to perceive that the finding of two mean pro¬ 
portionals is equivalent to increasing a given cube in a given ratio. In this way, 
his principal contribution lo the study of the cube-duplication problem follows 
from the same techniques which characterize his study of plane figures. This 
study of quadratures leads readily to the further question as lo whether a com¬ 
parable construction for the circle might be produced. Here, of course, (lie quest 
for a solution fails, but not before the intriguing detour into the construction and 
quadrature of the lunules. To be sure, only three cases arc actually found, 
although Hippocrates was surely aware of the defining conditions for an unlimited 
class of quadrable eases. The tantalizing result of having squared a certain lunulc 
in combination with a circle would of course deceive no one into supposing that 
the circle itself had been squared. But in all these instances an important principle 
has been established: that curvilinear figures, specifically those associated with 
circular arcs, are not different in kind from rectilinear figures as far as their 
quadrability is concerned. Thus, when the later commentator Ammonius (5ih 
century a.d.) presumes to have affirmed the impossibility of the circle quadrature 
through the intrinsic dissimilarity of the circular and the rectilinear, by analogy 
with the noncomparability of curvilinear and rectilinear angles, his disciple Sim¬ 
plicius brings forward the quadratures of the lunules as an immediate 
counterexample. 91 

In (he matter of the lunules, the standard assumption is that Hippocrates 
consciously restricted the means allowable in their construction lo the “Euclid¬ 
ean” devices of compass and straightedge. Although the eases he presents arc 
indeed constructiblc by these means, the view Mies in the face of his unexplicated 
introduction of a tieuxix in the construction of his third lunulc. While of course 
the compass and straightedge were by this time long familiar as constructing 
instruments, and while the range of constructions elTcctiblc by these means was 
doubtless already appreciated to be quite extensive, nevertheless other means 
were then known, and still others would later be invented. The enterprise of 
discovering the solutions to problems could hardly be well served by the im¬ 
position of such a restriction at this early stage. Hippocrates must surely have 
been interested in finding constructions, using whatever means were available 
lo him. The explicit restriction to one or another mode of construction is by its 
nature primarily a formal move, motivated by the urge to divide and classify 
the collected body of established results. Until the geometric corpus had attained 
a size and diversity meriting such efforts, there could hardly lie much sense in 
engaging in these formal inquiries. While that level surely was reached around 
the time of Apollonius, it is open to debate whether it had already reached that 
level at the lime of Euclid. 94 Applying this notion lo the much earlier time of 
Hippocrates must be viewed as purely anachronistic. 

There is a subtle distinction to he made here. Hie activity of seeking the 


solutions to geometric problems is intrinsically formal, to the extent the con¬ 
structions arc to he provided within an implied context that imposes some re¬ 
strictions on the available means and some measure of deductive ordering in the 
justifications. Thus, if one can rightly portray the most ancient traditions as 
empirical in that physical measures might be acceptable to justify certain claimed 
geometric results, then one cannot assign to these an involvement in problem 
solving. For in this environment any figure will be squarable, lor instance, to 
within any desired |>crccptil>lc degree of accuracy. By virtue of addressing the 
study of geometry in abstraction from overtly empirical measurements, the caily 
Greek geometers would soon come upon configurations whose production proved 
intractable. For instance, the range of techniques known for the construction and 
study of similar rectilinear plane figures would he recognized as leading to no 
evident solution lor the cube duplication or the circle quadrature. It is surely no 
mere coincidence, then, that Hippocrates was the first known syslematizer of 
geometry as well as the first to formulate these constructions as problems, while 
such an awareness among earlier geometers, even Ocnopides, is quite dubious. 

On the other hand, the conscious restriction to a specific set of constructing 
techniques, like the compass and straightedge in sharp separation from others, 
would be premature at Hippocrates* lime. This, I believe, may explain why the 
problems of rectifying and dividing circular arcs did not seem to attract interest 
until a century or more later. For a tradition of geometric practice which includes 
forms of protractors and cords among its tools might readily admit their abstract 
analogues in the more formalized study of figures, just as the use of sliding 
rulers gives rise to mouses ami (lie use of compasses suggests the Euclidean 
postulate ol circles.” Thus, the (riscction of the angle became a problem only 
after a lightening of the restrictions on construction techniques, and hence only 
within a formally more sophisticated geometric field. By contrast, the problems 
of the cube and the circle present difficulties at the much earlier level of devel¬ 
opment. Indeed, one may anticipate that il was through the investigation of these 
and related problems that the clearer conceptions of the general nature of the 
problem-solving enterprise were lo emerge. 

NOTES TO CHAPTER 2 

I’roclus, In hucHdem, cd. Ericdlctn, pp. 283. 333. On Ocnopides and Hippocrates, 
of. T. L. Ilcalli, History of Greek Mathematics, I, pp. 174-202 and ihe articles in Pauly 
Wissowa and Ihe Dictionary of Scientific Hiography. 

* Heath, op cit., p. 17b; cf. A. Szahrt, Atifon^e(tergriechischenMathematik, III. 19. 

I criticize this view in rny "On the Early History of Axiomatic*.” p. 150. 

’ G. Nciigchancr. Mathematische Keihchrift lexte, I {Quellen und Studien, 1935, 
3 : A), pp 137 142 On Ihe "rope stretchers" see Heath, op. at., pp. !2lff, and p. 
178, where the term appears in connection with the mathematical talents of IJcmocrilus. 

4 This view is advocated with particular zeal by A. Szabd, as in the work cited in 
note 2 aimvc. 

' Ocnopides is reputed, for instance, to have discovered the obliquity of the ecliptic; 




iCC Hcaih, op. at., p. 174. This is of course merely a testimony of llic infancy of 
astronomical science among ihc Greeks in the 5lh and 4th centuries, for such facts were 
certainly long familiar among ihe Mesopotamians. Of course, the specifically geometric 
approach to astronomy is characteristic of the Greek style, as contrasted with the arithmetic 
approach of the Mesopotamians. In this sense, the obliquity might indeed have been a 
new feature with Oenopidcs. 

6 Some of the instruments used in the practical work of geometry and architecture 
are named by Aristophanes (e g., Hints. 1001 If) and Plato (e.g., Philehus 51c, 56b, 
62b). Among later developments one may note the elaborate sighting device described 
by Hero (in his Dioptra, Opera III) and the compass for drawing the conic sections 
devised by Isidore of Miletus (see the interpolation into Eutocius' commentary on Ar¬ 
chimedes' Sphere and Cylinder . in Archimedes, Opera, cd. Heiberg, 2nd cd.. Ill, p. 
84), An instrument of the latter kind is described by al-Quhi and other Arabic geometers; 
see the account by F. Woepeke, l/Algehre d'Omar , 1851. p. 56n. In the same class may 
be mentioned the various mechanisms for the cube duplication used by “Plato," Era¬ 
tosthenes, and Nicomcdcs (cf. Eutocius, op. cit.. pp. 56 ff. 90 ff, 98 ff; these are discussed 
in Chapters 3 and 6). A survey of the devices familiar in the ancient practical geometry 
is given by 0. A. W. Dilke, The Homan Land Surveyors , Ch. 5. On the ancient archi¬ 
tectural instruments of stone dressing, in particular, the set square, compass, ruler, and 
plumb line, see A. Orlandos, Les Maifriaux de construction et la technique architectural 
des anciens Rrecs, Ecole fran^aisc d'Athfcncs; Travaux ft Mdnoires, 16, pt. 2, 1968, pp. 
59-69. 

’ See, for instance, Healh, op. at.. I, pp, 218 ff; Meeker. Mathematisches Penken 
der Antike, pp. 74 ff. 

* Theon, Expositor rerum ut ilium ad leyendum Platonem, cd. Hiller, p. 2. Plutarch, 
Moratia 386c. 579h. These passages have already been cited in Chapter 1. 

* Cf. Archimedes, Opera, III, pp. 88-96. 

w See, for instance. Heiberg's note in Archimedes. Ill, p. 89n; Healh. History . I. pp. 
244 f; I. Thomas, Greek Mathematical Works. I, p. 256n; B. L. van der Wacrdcn, Science 
Awakening, pp. 160 f. They all merely refer to von Wilamowitz as having established 
the inaulhcnticily of the document; see note 12 below. 

M Eutocius in Archimedes, III, pp. 88-90 (i), 90 (iil, 90-92 (iii). 92-94 (iv), 94*-96 
(v). 

** "Kin Wcihgeschcnk des Eratosthenes," (idttmyrr Nachrichten, 1894; repr. in Kteiuc 
Schn/len II. 1971. pp. 48 70. 

•* Collection, cd. Hultsch (111) I, pp. 56-58. 

M Eutocius, op. cit ., pp- 90, 96. 

Sec Pig. I ta b) in Chapter 6 for Enitosthcncs' consmiclion. 

** Eutocius, op. at., pp. 90, 96. 

11 For a gauge of the longevity ol such a structure, one may consider the grave marker 
of Archimedes, all hut lost within a century and a half of its foundation; the circumstances 
of Cicero's rediscovery of Ihc monument in Ihc 1st century n r arc related in his Ttisculan 
Disputations. V, 64ff. 

'• Recall that Hypatia, the learned daughter nf Thcon of Alexandria, died in 415 a r> 
at the hands of a mob of rioting monks; cf. Heath, History. II, pp. 528 f, and P. Brown, 
The World of luite Antiquity . Ijondoii. 1971. pp. 103 f. 


See A. Fitzgerald, The i.etter\ of Sy nest us, pp. 258-266 (csp. pp. 262-266); 
Fitzgerald provides a useful discussion of Syncsius* career in the Introduction and notes. 

'* Cf., in particular, the prefaces to Sphere and Cylinder 11 and to Conoids and 
Spheroids', those to Quadrature of the Parabola. Sphere and Cylinder I and Spiral Lines 
provide a certain small amount of contextual information. 

m Indeed, G. J. Tooiner expresses misgivings over the general assumption that Al¬ 
exandria was a monopolizing center of technical studies in this period; cf. his Diodes . 
p. 2. One notes that the prefaces to works by Diodes and by Apollonius arc somewhat 
more ample in background information than those to the Archimedean writings. 

99 Opera. II, pp. 216-258. For accounts, sec the Archimedes monographs hy Heath, 
by Dijkstcrhuis, and by Schneider. 

11 In the 43 pages of the standard edition of the SatuLReckoner, one finds about a 
dozen second-person forms (e.g., pp. 216. 218. 220. 234. 244. 246. 258); in Eutocius* 
text of Eratosthenes, a five-page document, there arc three (pp. 90. 94. 96). not counting 
the epigram. 

n For an account of the general background, see M. 1. Finley, Ancient Sicily , New 
York, 1968, Ch. 9. I apply these materials toward the dating of the Sand‘Reckoner in 
my “Archimedes and the Elements." pp. 234-238. 

M Eutocius, op. at., pp. 88 90; cf. Heath. History , I. p. 245. 1 have inserted the 
reference letters. 

*' On Arehyms, see Heath, op. cit., pp. 213-216, 

* Euiocius. op, cit., p. 84. 

f ’ Ibid . p. 96 

** Expositio, cd Hiller, p. 2. 

* Moratia, 386c, 579b-d. Cf. note 8 above. 

* Ibid,, 718c f; Vita MarceRi xiv, 5. Ihc latter contains an extended account of the 
life and exploits of Archimedes: see the Archimedes surveys cited in note 21 above. 

M Details appear in Chapter 3. 

w Heath. History, I, p. 2*16; van tin Wneidcn, op. at., pp. 161-163. The Inner seems 
to treat the story ns a dramatization by Eratosthenes; hut in a more recent effort, he wishes 
to trent it as a serious datum reflecting a ritual tradition within early mathematics; see 
his “On Pre-Babylonian Mathematics (II)." Archive for History of Exact Sciences, 1980. 
23. pp. 37 f. This thesis is advocated hy A. Scidcnbcrg in several articles, particularly. 
“The Ritual Origin of Geometry.** Archive for History of Exact Sciences, 1963. I. pp. 
488-527, 

M Euiocius, op. cit., p. 88; to he discussed further below. 

14 Note in particular Plato's criticisms of geometric research and his recommendations 
for mathematical education in Republic, Hooks VI-VII. 

M Cf. Post. Ana. \, 7 and Meta M.3. Passages on the ordering ami nature of the 
sciences arc collected ami discussed by Heath, Mathematics in Aristotle, pp, 4-12, 46, 
59 f, 64-67, 225. 

* Eutocius. op. cit., p. 88. 

” Ibid., pp. 88 90. 






M A form of (he proof, applied to the mure general ease of similar solids, appears in 
Euclid** Elements XI, 33 (purism). 

* hi Euclidem. p. 213. On the method of analysis, see Chapter 3. 

40 De Anima II, 2. 413 a 16-20. This passage is noted, if briefly, in most editions of 
the work; cf. W. D. Ross, Aristotle's De Anima, Oxford. 1961, p. 217, and also Heath, 
Mathematics in Aristotle , pp. 191-193. Most commentators, however, attempt to construe 
it as a specific reference to the Euclidean const met ions of either or both of the two 
problems, rather than as a statement of the reduction of the one to the other. 

*' This fragment, preserved in Simplicius' commentary on Aristotle’s Physics , is 
discussed further in this ami the next chapter. 

47 On the ancient Egyptian and Mesopotamian mathematical traditions, see O. Neu- 
gebauer. Exact Sciences in Antiquity . Ch. 2, 4; and van tier Wacrdcn, op. cit ., Ch. 1, 
2. On the thesis of the ritual origins of technique, see note 32 above. Establishing a 
perspective on the relation of geometry and philosophy in the pre-Euclidean period is an 
atm of Chapter 3 below. 

41 In Euclidem, pp. 422 f. Proclus cites as an example of a theorem on the circle 
quadrature Archimedes' Dimension of the Circle, Prop. I; see Chapter 5. 

44 Problem 48; cf. the edition by A. Chaco et of.. 1927 *29, II. Plate 70. For a 
discussion, see R. Gillings. Mathematics in the Time of the Pharaohs, 1972, pp. 139- 
146, and H. Engels. “Quadrature of the Circle in Ancient Egypt.** Historia Mathematica, 
1977, 4, pp. 137-140. The rule is equivalent lo approximating it as 256/81, that is, 3 
13/81 or a hit less than 3 1/6 

4 * That this relation was recognized is indicated by the use of the correct rule for the 
volume of the truncated pyramid in the Moscow Papyrus tc. 1600 im‘); see van dcr 
Wacrdcn, op. cit. , pp. 34 f. For this would seem to require knowledge of the t : 3 ratio 
between the pyramid and the corresponding parallelepiped. Physical procedures for the 
measurement of irregular surfaces and solids are found in the (iicck metrical tradition: 
cf. Hero, Metrica I, 39 and II. 20. 

** The Birds, lin. 1001-1005; cf. Heath, History, l, pp. 220 f and I. Thomas, op. 
cit., pp. 308 f. Thomas translates kampxtos kandn (literally, “curved ruler") as “flexible 
rod," and this may be supported in the light of Aristotle's mention of (he “leaden rule" 
used by architects (MV. Eth. 1137 b 30) and by Diodes* use of a flexible ruler of horn 
for the purposes of curve plotting; see loonier. Diodes, pp. 159 I . H. Mendel! informs 
me of an alternative rendering of the passage which conforms better both to sense ami 
to the standard punctuation: “If I lay out the ruler and insert this curved compass from 
above—do you see?... I shall measure with the straight niter by laying (it) out..." In 
this way, “curved" iiunlitics “compass** (cf. Aristophanes, Clouds , 178) rather than 
“ruler**. This of course docs not affect the later testimonia to the use of flexible rulers 
in geometric practice. 

41 Cf. Heath, he. cit. The passage itself gives no fully clear notion of just what Meton 
is up to. and perhaps this is pan of the joke. 

w Physics 185 a 16; Sophistical Refutations 171 b 15; cf. Heath, Mathematics in 
Aristotle , pp. 33-36. 

44 Simplicius, In Physica, ed, Diels. 1, p. 60. The fragment from Eudemus occupies 
pp. 61-68, followed by an assessment on pp. 68 f; Simplicius lakes up other versions 
of the circle quadrature on pp. 53 -60. 


Collection (V), I, pp. 340 342. See my discussion in “Infinity and Continuity.** 
pp. 127 ff. 

M On (his terminology, see my “Archimedes and the Elements ,** pp. 240n. 254n, 
264; and iny *'Archimedes and the IVc-Euclidean Proportion Theory,** p. 196. 

On the Endoxcan methods and their attribution via Archimedes* Quadrature of the 
Parabola and Sphere and Cylinder I, see my “Archimedes and the Pre*Euclidean Pro¬ 
portion Theory." pp, PM If. 

For a general account of Antiphon's thought, see W. K. C. Guthrie, A History of 
Creek Philosophy , III, pp. 285 *294. Simplicius' passage on his circle quadrature [In 
Phy.uea, pp. 54 ft would appear lo derive from Eudemus. for his specific critique of this 
argument is cited (ibid ., p. 55). The same critique, if without express citation of Eudemus, 
is given also by Themistius (4th century a n . In Phyxica . ed. H. Schenkl, LAG 5. Pt. 
II, 19(H), pp. 3 f), namely, that it violates the principle of the unlimited divisibility of 
continuous magnitude. Hut Simplicius also cites an alternative critique by Alexander, so 
that he may well have derived his account of Antiphon via Alexander, rather than directly 
from Eudemus. 

u Simplicius, op. cit.. pp. 54 f. 

Ibid., p, 55 fscc note 53 above!. On Protagoras’ treatment of this issue, see Ar¬ 
istotle's Metaphysics II, 2. 997 b 32 and its discussion by Guthrie, op. at.. II. p. 486 
and IK. p. 267. On Democritus' work, see Heath. History, 1. pp. 176-181 and Guthrie, 
op. cit II, Ch. viii, especially pp. 487 f. 

* Cf. Eudemus * view, cited in note 53 above. 

” See, foi instance. Heath. History. I, p. 224. 

See my “Infinity and Continuity." pp. M3 I. The stimniatinit theorem for the liuitc 
case is given in Elements V. 12. 

Hippocrates seems to appeal to a concept of proper “parts" as the basis of his 
notion of ratio; cl. Simplicius, op. at., p. 61 and the discussion by Heath. History. I. 
pp. 187-191. I bis could provide the precedent for a theory of proportions lor commen¬ 
surable magnitudes; but the extension to incommensurable magnitudes, as in the Etidoxean 
theory, had not yet Ih*c»» worked out at Hipocrates* time. See my “Archimedes and the 
Pre-Euclidean Prnpnilinn Theory." ami Chaplet 3. 

Enrlher examples are found in Pappus' measurements of figures bounded by spirals; 
see Chapter 5. 

Plutarch. M or alia 607 f (cf. Diels and Kranz, A 38, II, p. 14). The view that this 
might signify a writing by Anaxagoras on circle quadrature is discounted by J. Burnet 
(Early Greek Philosophy. 4lli ed.. p. 257) and by Guthrie (op. cit., II. p. 270). 

In Euclidem . pp. 65 f. Vitruvius also mentions Anaxagoras for a study of per¬ 
spective. cl. Heath. History, I. pp. 172-174. For a general account of Anaxagoras, see 
Guthrie, op. at.. II, Ch. iv. 

Prior Analytics II. 25; cf. the passages ciietl in note 48 above. 

M See note 49 above. 

M Simplicius, op. cit., pp. 56 f. 

™ readily suggests an alternative construction in which the right triangle may be 
scalene rather than isosceles. Here ton the two lunulcs will sum to the area of the triangle. 


**_r », ♦ • 



c 



Figure 8 


Simplicius* text docs not mention this variant, but it lias slipped into modem accounts 
as if by Hippocrates himself. M. Simon notes having failed to discover this variant in 
works prior to the late 17th century (“Lunulac Hippocratis,” Archivfur Mathematik toui 
Physik. 1904-05, H, p. 269). But a medieval precedent may be found in ticalincnts by 
ibn al-Haylham and Joannes de Muris; cf. M. Clagett, Archimedes in the Middle Ages, 
111, pp. I3l5n, 132If. One notes that since the small segments over the sides AC, CB 
are not similar, neither are the lunules; hence the lunules will not be in the ratio of the 
similar triangles ACD, DCB, respectively, but rather, lunulc AC will be less and lunulc 
CB greater than its corresponding triangle. Ibn al-Haytham observes that the lunules will 
be equal to the respective triangles ACE, ECB (where \i is the center of the circle) when 
the arcs AC ami CB are equal. The relation is of course not general, for in the scalene 
case the lunulc over (lie greater leg will he greater than that over the smaller leg; the 
maximal discrepancy occurs for central angle CEB having sine equal to 2/ir (approx. 
39'/i°), where the lunules will be in the ratio 3.1616... : I. (This is the configuration 
shown in Fig. 8.| 

w For a review of modem interpretations, particularly those of Tannery and Bjdrnbo, 
see Heath, History, I, p. I96n and Thomas, op. < i/., I, p. 3H)n. The opinion of Alexander 
is cited by Simplicius, op . at.. pp. 57 f. 60, while his own view appears on pp. 60. 69. 

M For full accounts, see Heath. History , I. pp. 191-200 and Thomas, op. t'it.. I. pp. 
234-253. A major interpretive issue here centers on the segregation of Eudcmiis* text 
from Simplicius* interpolations. Proposals have been offered by Allman, Diels (IKK2), 
Tannery (1883), Rudio (1907), and Becker (1936). For references, see Thomas, op. at.. 
p. 237n. Rudio's translation and commentary contain many useful insights ("Bench! des 
Simplicius... ."^Bibliotheca Mathematical 3„ 1903, pp. 7-62); see also my “Infinity 
and Continuity.” pp. 127-130. Thomas follows Kudio*s abridgment, so that the reader 
interested in judging for himself must go back to Diels* text. 

m This is done by applying the principle that in right triangles the square on the longest 
side equals the sum of those on the other two sides, but in obtuse triangles it is greater 
and in acute ones less; cf. Elements I, 47 and II, 12-13. In the present ease, since angle 
BAG is obtuse. BG 2 > BA 2 + AG 7 - 2 CiD\ whence BD 2 (^ 3 C.lV) < BG 2 T GJ) 2 . 
so that angle BGD is aculc. Hence, arc BGD is greater than a semicircle, since the angles 
inscribed in segments arc proportional to the corresponding supplementary arcs (cf. 
Simplicius, op. at., p. 60). 


Specifically, this text includes archaic usage of the form “the line on which AB,“ 
where later writers would invariably say “the line AB”; see below, note 90. 

Signs of such minor textual corruption appear elsewhere, for instance, in the treat¬ 
ment of case (iii) where certain steps in the construction are out of the proper order 
(Simplicius, op. at., p. 65); cf. Thomas, op. rr/.. p. 244n. 

On the method of analysis and synthesis, standard in the later problem-solving 
tradition, see Chapter 3. 

" For if one introduces line 1IZK, the angles at K and B will be equal; hence. Z is 
the vertex of the isosceles triangle ZKB. 

24 Simplicius, op. at., p. 64. 

See, in particular. Chapters 5. 6, and 7. A survey of ancient tteuses is given by R. 
Boker in Patdy Wis.sown, Suppl. ix. 1962. col. 415-461. Cf. note 86 below. 

** Simplicius, op. at ., p. 66. The text adds, superfluously, “and the rectilinear figure 
is [the meniscus | with the two segments, but without the three.” 

” /hid., p. 67. Writing “|hc squared)*' after “seeing that” (or “since in fact,” in 
my rendering of eiper Am). Heath assumes an ellipsis, and he is well supported in view 
of the grammatical role ol eiper as a subordinating conjunction. Iliis entails, however, 
the fallacy of concluding the general from the particular Wishing to avoid this. Tannery 
and Bjdrnbo have proposed excising “every” and “seeing that** as interpolations (see 
Heath, he. cit.). 

n Simplicius, op. t it., p. 68. 

/hid ., p. 69; cf. Aiisiollc, Physics 185 a 16 (note 48 above). 

Wl Note that lines 0C and d'C* w ill be tangent to the circular arc. 

This is under the restriction that 0 lie less than 180/m, for m the number of elements 
in the polygonal arc. 

" 2 Cf. Heath. History. I, p, 200. 

M E. Landau, “Ueber qnadrierbare Krcisbogcnzwcieckc.” Sitzun$sberichte. Berlin 
Math. Gcs.. 1902-03, pp. 1- 6 (suppl. to Arehivfiir Math. tau/Phys.. 4, 1902 *03); cited 
by I\ Enriques (ed.). Frozen dvr Llementar^eometrie, 1923. II, pp. 304-308. Tschak- 
alolf, **Beitrag zum Problem del quadrierbaren Krcisbogcii/.wciccke,” Math. Zeitschrift. 
1929, 30, pp. 552 W; this ami other efforts are cited by L. Biebcrbaeli (cf. note 89 
Mow), pp M0 f, 159 

114 References arc given by A. I). Steele, “Zirkcl und Lineal,** Quei/en ton! Stttdien. 
1936, 3 : II. p. 317. Hie algebraic condition to which the ease (1,9) reduces has some 
consinictihlc roots, but these arc imaginary; its real roots arc nonconstructiblc. 

' ^ce Heath, futclid, I, pp. 386 f and Chapter 3 below. 

“ Steele provides an alternative planar construction of this neusis. following the model 
of a result from Apollonius* He uses as discussed by Pappus [op. cit.. pp. 319-322). But 
Steele goes on to insist, on textual grounds, that the Hippocrates-Eudemus passage, as 
extant, could not have intended any such alternative method in substitution for the neusis. 

See Heath, Archimedes, pp. ci-ciii, who follows Zeuthen in supposing that Ar¬ 
chimedes must have understood alternative constructions of the But Dijkstcrhuis 

{Archimedes, pp. 138 0 insists, I believe correctly, that Archimedes accepted die names 
in their own right; see Chapter 5. 


“AM) continually decreases fiom IWI/(m_* »») Jo 0. the ratio X : Y continually 
increases front I : I to m : n. Since lire ratio v'm : Vn lies within this range, X : Y will 
assume this value for some angle within the corresponding domain of 0. One may observe 
that Hippocrates’ neusis for the case (2. 3), when X = Y. yields a construction for the 
regular pentagon. Some have wished to assign a method of this sort to the ancient 
Pythagoreans; but. unfortunately, there is no documentary support for this claim (cf. R. 
Bolter, “Winkcltcilung." pp. 137 ft. 

* In this connection. Steele notes Viola’s treatment »>r the ease (I. 4); sec op. til.. 
p. 3IK. Nernrs may be used for the solution ol thiid-ouler problems like the cube 
duplication and the angle bisection; sec Chapters 5. A. and 7. hit an account of this 
construction technique, see L. Biebcrbach. Throne tier K eomeinx,hen Konxirukiionen. 
Basel. 1952. Ch. 16. 17. 

* Sec note 70 above. The language of proportions here is also archaic. Hudemus - 
text speaks of the proportion A : B * (i: I) by the expression A is greater than H by 
the same pari that Ci is greater than IV (Simplicius, op fit.. p. 60). This may be compared 
with the phrasing in Arehy.as (fr. 2; Dicls-Kratu. 47 I) 2.1. PP . 435 0 and in the pseudo- 
Aristotelian Met hanks. Ch 20. By contrast, line lid. Archimedes, and other late writers 
will say "A has the same ratio to B that Ci has to I).” tn more simply. "A is to B as 
G is to D.** 

« Note, for instance, that the commentators commonly speak of "those in the circle 
of..." (hoi peri), thus leaving it unclear whether they refer to the master of the tradition 
or to his followers, (iuclcmiis himself was evidently willing sometimes to make historical 
claims on the basis of inference. He says of Thales, for instance, that he knew a certain 

theorem on congnicnl triangles Mem. I. 26). foi lie "mtisl have used this" Mi detenu.. 

the distance of ships at sea. as lie is reported to have done tef I'mclus. In l-.m Intern, p 
352). The difficulty of discerning the actual basis of this and oilier claims by l-iidcmos 
has thus led to a spectrum of widely divergent views on the achievements of llialcs. l-nt 
references, see W. Burken. Tore ami Sneme in Ancient ryiltnitorranism. 1972. pp. 
415-417. 

m * Proclus. tn EuvUdem. p. 66. 

Simplicius, op. at., pp. 59 f. Ammonium here refers to ihc “horn angle** clcfliiicd 
by the space between the circle and its rangem (cl. klvni. Ill, 16 anil Heath, kinluL II. 
pp. 39 ff). Proclus makes several references to this and related mixed and curvilinear 
angles (cf. In Emt. pp. 122, 127, 134k These ligurc within Philnpnmis* discussion of 
Bryson*s circle quadrature (see Chapter 3). 

** This will be discussed in more detail in Chapters 7 and K. 

" In particular, the availability of “curved (flexible) rulers** suggests an extremely 
simple practical procedure for these constructions; sec note 46 above and Chapter 3. 


t 


CHAPTER 3 


The 
Geometers 
in Plato’s 
Academy 

Through Hippocrates* efforts a start was made inward the investigation of geo¬ 
metric problems liul tlu: first discovery ol actual solutions lor more difficult 
problems, like the circle qiiadrnliiic and the cube duplication, was made by 
geometers in the genenuions alter him. 1 Uppocratcs* reduction of the latter 
problem, lor instance, to the form of Finding two mean proportionals between 
two given lines served as the basis for the successful constructions by Archytas. 
Eudoxus, ami Mcnaechnius. At the same time, new geometric methods were 
introduced, in particular the use of special curves generated through the sectioning 
of solids or through the geometric conception of mechanical motions. Purther- 
more, Hippocrates* precedeni for the “reduction** of one problem to another 
could lead others to the articulation of the versatile technique of geometric 
“analysis.** 

During the dlh century it Plato's Academy in Athens became a center for 
geometric studies. This is reflected, as we have seen, in such legendary accounts 
as that of the Delian made, pnrliaying Plato’s interaction with mathematicians. 
Although later authorities sometimes assign to him the discovery of specific 
technical results, like a method for the duplication of the cube, the veracity of 
such icports is dubious and generally discounted. In (he tcasonablc assessment 
by Proclus, it was rather through the special position he afforded mathematical 
study within his program of philosophical education, elaborating a plan initiated 
by the older Pythagoreans, that Plato encouraged mathematical research, as well 
as through his incorporation of technical examples into his writings to illustrate 
points of method. 1 Platt* appears tn have been much impressed by the technical 
rigor of some of the older geometers, notably Thcodorus of Cyrcnc, a content- 




50 Ancient Tradition of Geometric Problems 


The Geometers in Plato's Academy 51 


porary of Socrates, and by the genial insights of geometers of his own generation, 
in particular the Pythagorean Archytas of Tarentum and Theaetetus of Athens. 3 
Of other geometers, like Leon and Leodamas, we know little. But towards the 
middle of the century, Eudoxus of Cnidus became affiliated with the school, 
and he and his immediate disciples, like the brothers Menaechmus and Dinos¬ 
tratus, profoundly stimulated research into mathematics and mathematical sci¬ 
ence. It is natural to suppose that the formal precision characteristic of the 
geometric methods of Eudoxus developed in part as a response to the intellectual 
emphases of this philosophical environment. Interestingly, Eudoxus 4 tenure at 
the Academy coincided with that of the young Aristotle, so that the work of this 
geometer can hardly but have had its influence on the elaboration of the philos¬ 
opher's views on logic and the formal structure of science.' 

Recognizing these aspects of the context of research in this period will prepare 
us for its distinctive character. Even as the geometric field expanded impressively 
in the range of established technical results, there advanced parallel to this a 
rigorization of the forms of mathematical exposition. The latter has often been 
exploited as evidence of the influence of philosophical developments on math¬ 
ematical research and has induced many scholars to propose dialectical reasons 
for explaining technical issues, like the ancient treatment of infinity or the se¬ 
lection of methods of construction. But such views are too readily exaggerated. 
Surely, the technical disciplines were on the whole autonomous in setting the 
directions of research. If in this period geometry and philosophy interacted to a 
greater degree than at other times in antiquity, the effect on geometry was subtle 
and not specific: an enhanced sensitivity to questions of formal proof. By contrast, 
the reciprocal effect on philosophy could often be specific, as mathematics offered 
clear models for inquiries into the general structure of knowledge. Failure to 
appreciate this distinction must inevitably hinder the effort to capture the nature 
and motives of technical work in the pre-Euclidean generations. 

SOLUTIONS TO THE CUBE DUPLICATION 

According to Eratosthenes, three of Plato’s associates worked out solutions for 
the Delian problem: Archytas “via semicylinders,” Eudoxus “via curved lines,” 
and Menaechmus “via the triads cut out from the cone.” 4 Among the dozen 
methods in his survey, Eutocius presents some remarks on a text of Eudoxus’ 
method, along with complete details of the construction and proof of those of 
Archytas and Menaechmus, the former based on an account by Eudemus. In the 
present section we review these, adding a fourth method transmitted by Eutocius 
under the name of Plato. 

The construction of Archytas is a stunning tour de force of stereometric 
insight. 5 Its essential idea, following the form of the reduction of the problem 
by Hippocrates, is to find the two mean proportional lines AI, AK between two 
given lines AM, AD via an arrangement of similar right triangles as shown in 
Fig. 1. Archytas determines point K as the intersection of three solids. Starting 
with a semicircle whose diameter AD' equals the larger given length and in 




which the chord AB equals the smaller, he conceives an equal semicircle erected 
in the plane perpendicular to the given semicircle; as the diameter AD* turns on 
the pivot A, this circle will generate the surface of a torus (see Fig. 2). If a right 
scmicylindcr has been erected over semicircle ABD' as its base, then the torus 
will meet it along a certain curve. At the same time, he conceives that the given 
semicircle rotates about its diameter AD as an axis, so that the extended chord 
AB generates the surface of a cone. Let this intersect the “cylindric curve at 
K. Archytas then shows that this effects the construction: if one draws triangle 
AKD, the line KI perpendicular to AD, the line AM equal to AB, and the line 
MO perpendicular to AD, he can prove that the angle AMI is a right angle. 
Hence, the three triangles AMI, AIK. AKD arc similar, so that the four lines 
AM, AI, AK, AD are in continued proportion. 

The synthetic expository mode here adopted by Eutocius inevitably obscures 
the underlying pattern of thought. But an analysis along the following lines may 
suggest an approach available to Archytas. Since the objective is to find an 
arrangement of the triangle AKD such that AM has the given length, let the 
chord AK be conceived to rotate on A, and for each position of K let there be 
drawn KI perpendicular to AD and 1M perpendicular to AK (see Fig. 3). Then 
M will trace a curve such that the ray AM continually decreases from its initial 
value AD. assuming arbitrarily small values in the vicinity of A. At some 
position, then, AM will equal the given line AB, and from this the solution of 
the problem follows.* Instead of effecting this via a pointwise construction of 
the locus of M, one can seek the common intersection of the loci associated 
with the points K, 1, and M. 7 The locus of K. as the vertex of a right triangle 
of hypotenuse AD, is secured by keeping it on the circumference of the circle 
AKD (sec Fig. 4). The length AM can be kept equal to AB by setting M on the 
trace of B as its semicircle rotates on its axis AD. As for the locus of I. this 
can be solved as follows: in the desired terminal position, we require that IM 
be perpendicular to AK, where I is figured as the foot of the perpendicular from 
K to AD. Now, M lies on the circle ZMB of diameter ZB; setting 0 as the 



52 Ancient Tradition of Geometric Problems 



/ /' D 


Figure 3 

intersection of ZB with Al, MO is then perpendicular to both of those lines (for 
it lies along the intersection of the two semicircles AKD, ZMB, each perpen¬ 
dicular to the base plane which contains the lines AD, ZB). Since angle ZMB 
is inscribed in a semicircle, MO 2 = Z0 • 0B; since it is required by hypothesis 
that angle AMI be a right angle. MO 2 = AO • 01. Thus, A0 • 01 = Z0 • 0B, so 
that ZOB and A0I will be intersecting chords in the same circle (cf. Elements 
III, 35); that is, I lies on the circle ZABD'. Hence, one may take the semicircle 
ABD' as the locus of 1, so that the vertical KI will lie in the right semicylinder 
erected over ABD' as its base. 

Through an analysis of the sort just given, one sees that the desired config¬ 
uration of similar right triangles can be arranged through the determination of 
K as the intersection of the torus, semicylinder, and cone. The construction and 
proof by Archytas, as transmitted by Eutocius, follows precisely as the synthesis 
answering to this analysis. 

Eutocius possessed a text purporting to set out Eudoxus’ construction of the 
problem, but expresses reservations concerning its authenticity: 

Wc have happened upon writings of many clever men reporting on this problem 

|of finding the two mean proportionals). Of these, we deprecated the writing of 



Figure 4 


The Geometers in Plato’s Academy 53 



Figure 5 


Eudoxus of Cnidus, since he says in the introduction that he discovered it by means 
of curved ( kampylai ) lines, but in the proof, in addition to not using curved lines, 
he even finds a discrete proportion and then uses it as if it were continuous. This 
would be impossible to imagine in the instance of anyone even moderately versed 
in geometry, let alone in the instance of Eudoxus.* 

Scholars have understandably deplored Eutocius’ decision to omit any further 
details on this method, for one would surely expect that even this defective text 
preserved the basis for a satisfactory reconstruction. M Despite this omission, 
several proposals have been made. One makes use of the curve we mentioned 
above in connection with the analysis leading to Archytas’ solution (cf. Fig. 3). 
A second suggestion, advocated by Tannery, also starts from Archytas 1 figure. 10 
Tannery proposes that Eudoxus considered the orthogonal projection of the curves 
of intersection onto the base plane. The section of the cylinder by the torus 
projects onto the semicircle ABD'. The section of the cone and the torus gives 
rise to a plane curve of the following description (see Fig. 5): in the semicircle 
ABD' draw BN perpendicular to AD' and let an arbitrary ray from A meet BN 
at R; mark S on AD' such that AS = AR and let the perpendicular to AD' at 
S meet AR extended at P. Then the locus of P associated with all such rays is 
the second curve of projection, and its intersection with the semicircle ABD' is 
the point I which solves the problem. Although Tannery depends on the use of 
modem coordinate representations to display this curve, it can be worked out 
according to ancient geometrical methods without undue difficulty [see Fig. 6]. 
For consider the point Q on the intersection of the torus and the cone. From the 
torus, it follows that AQ is the mean proportional of AP, AD, since it is a chord 
in the semicircle AQD. From the cone, it follows that AQ = AT. Now, AB is 
a chord in the Semicircle ABD', so that it is the mean proportional of AN, AD'. 
Thus, AT 2 : AB 2 = AP • AD : AN • AD' = AP:AN (since AD = AD'). By 
similar triangles, AT : AB = AS : AN, whence AS 2 : AN 2 = AP : AN, or AS 





54 Ancicni Tradition of Geometric Problems 


A 


D 


(a) 


(b) 

Figure 6 




: AN = AP : AS. But by similar triangles, AP : AS = AR : AN, so that AR 
: AN = AS : AN, or AR = AS. This last result leads to the construction of 
the curve as specified above, where one must introduce some form of point wise 
procedure actually to produce the curve and find its intersection with the semi¬ 
circle ABD'. 11 

A proposal of quite a different sort has been made by R. Riddell, who links 
the recursive configuration of right triangles essential to Archytas’ method with 
a kinematic configuration derived from Eudoxus’ work in geometric astronomy. 12 
Addressing the problems of describing the motion of planets by means of uniform 
circular motions, Eudoxus hit upon the conception of the “hippopede” curve: 
if one considers a point P set on the equator of a rotating sphere, and one embeds 
this Sphere within a second sphere rotating with an equal speed in the opposite 
direction and communicating its own motion to the inner sphere, then if the two 
spheres have the same poles, the point P will merely remain fixed; but if the 
spheres are inclined at an angle, P will trace a curve in the form of a double 
loop, or figure-8, for each complete revolution of the two spheres. (See Fig. 
7a.) It was named “hippopede," or “horse fetter,” for its shape, and may be 
alternatively conceived as the intersection of the inner sphere with a right cyl¬ 
inder. 11 Projected onto the equatorial plane of the outer sphere, the curve becomes 
the circle ARH (see Fig. 7(b)). The effect of the rotation of the inner sphere is 
to raise P above this plane from its initial position at A, so that its projected 
position is on the circle QR'G; the effect of the rotation of the outer sphere is 
to transfer the whole figure back over its initial position ARH, thus leaving the 
projected position of P at R. 14 If this looping path of P is now superposed over 
the slow rotation of a point on the equator of yet another enclosing sphere 
(corresponding to the ecliptic circle on the celestial sphere), then the resultant 


The Geometers in Plato's Academy 55 



motion of P gives a fair semblance of the path which planets are seen to mark 
off in the heavens. Unfortunately, this model proves remarkably inept at saving 
the observed appearances of planetary motions: it fails to account for the vari¬ 
ations in orientation, size, and shape of successive loops; in some instances, 
like Venus, it does not yield a retrograde motion at all; and it places the planet 
always at the same distance from the observer and so leaves unexplained the 
differences in brightness. 13 Surely, such difficulties as these induced later as¬ 
tronomers to try entirely different approaches, like the heliocentric model of 
Aristarchus, setting the planets each in its own circular orbit around the sun and 
each with its own period of revolution. 

But if the Eudoxean scheme of homocentric spheres held limited interest for 
astronomers beyond a few decades after Eudoxus, this need not signify that the 
curve he introduced failed to draw the attention of geometers. Indeed, the hip- 
popede is mentioned several times by later authors for one or another of its 
features. 16 Riddell thus invites us to consider another curve generated via a 
modification of Eudoxus’ arrangement. Instead of having the outer sphere rotate 
in reverse with the same speed as the inner, let it rotate with twice that speed. 
This will have the effect of returning the projected position of QR'G through 
an equal angle to the opposite side of its initial position, so.assuming the position 
JRH (see Fig. 8(a)). The space curve traced by P will thus arch to an upward 
maximum corresponding to the first quarter-turn of the inner sphere, then move 
downward, arriving at its minimum at the end of the third quarter-turn. 17 The 
curve of projection will accordingly pass through two in ward-facing arches, as 
it weaves within the space bounded by an outer circle and an inner ellipse (see 
Fig. 8(b)]. Now consider the projection R of an arbitrary position of P [see Fig. 



The Geometers in Plato's Academy 57 


8(c)]. Draw the line RT perpendicular to the initial radius OA, and meeting the 
terminal radius OJ at S. Since the angles at 0 are equal, JQ is perpendicular to 
OA, meeting it at V. Drawing SV, Riddell shows that VS will be perpendicular 
to OJ on the condition that OH is one-half OA. 1 * For this makes R' the bisector 
of QV, whence JK (side of the rhombus RSKJ, where RJ = R'Q) will equal 
one-half JV; hence, VK = KJ - KS, so that angle JSV is a right angle. Thus, 
the triangles OTS, OSV, OVJ fall into the configuration central for Archytas* 
duplication of the cube: that is, the lines OS, OV are two mean proportionals 
between OT, OJ. Thus, to find the two means between two given lines, one 
produces the projection of this modified Eudoxean curve, where OA equals the 
larger given line and OH equals onc-half of the line; marking off the shorter line 
as OT, one then draws the perpendicular to OA at T, meeting the curve at R, 
and one locates the point R' as the projection of the point P' where the circle 
through P parallel to the equator of the outer sphere meets the equator of the 
inner sphere. Then, the perpendicular to OA from R' meets the outer circle at J 
and OA at V, from which the diagram may be completed to yield OS, OV as the 
required mean proportionals. 1 v 

Relative to these proposals by Tannery and Riddell, it is a mere quibble to 
point out that Archytas does not actually introduce the section of the torus and 
the cone, nor does the ancient tradition ever signify such a variant on Eudoxus* 
hippopede, 20 These are both ingenious constructions, quite in the spirit of the 
ancient geometrical methods. It will emerge from what follows, however, that 
further considerations point to a different alternative. 

The construction “according to Plato” is the first of Eutocius* texts on the 
cube duplication, and thus follows immediately his brief comment on the omitted 
Eudoxean method. 21 The “Platonic” procedure employs a mechanical device to 
produce a configuration consisting of three consecutive similar right triangles 
GBD, DBE, EBA, where the sides AB, BG are given (see Fig. 9(a)]. Its base 
is the beam HO pivoting on the given point G such that its endpoint H always 
lies along the line of AB [see Fig. 9(b)]. A second beam HZ is fixed at a right 
angle to the first. A third beam KL is free to slide along HZ, but is always 
perpendicular to it. If K is set along the line of GB, then KL will meet BA (or 
its extension), say at A', and when this point comes to coincide with A in the 
course of the device's manipulation, the construction will have been effected, 
so that DB, BE will be two mean proportional lines between the given lengths 
GB, BA [see Fig. 9(c)]. 

The Platonic provenance of this method is open to serious question. First, 
Eratosthenes makes no reference to it among his remarks on the early efforts at 
cube duplication. This is all the more noteworthy since his interest in the Platonic 
philosophy is evident through his writing of the Ptatonicus and his elaboration 
there of the story of Plato's involvement with the Delian oracle. 22 Surely, then, 
Eratosthenes would have been particularly keen to indicate an actual solution by 
Plato of the Delian problem, had he known of one. Furthermore, this method 
depends on the conception of a mechanical device. Indeed, Eutocius* text goes 
into considerable detail on the fashioning of the grooves and the pivots ensuring 
that the beams maintain their perpendicular orientations throughout their motion. 


i 







58 Ancient Tradition of Geometric Problems 


The Geometers in Plato's Academy 59 


z 




One is astounded at the flexibility of the traditions which on the one hand attribute 
such a mechanism to Plato, yet on the other hand portray him as the defender 
of the purity of geometry and the sharp critic of his colleagues for their use of 
mechanical procedures in geometric studies. But surely the latter tradition, as 
typified in Plutarch’s version of the Delian story, is closer to the mark in capturing 
Plato’s own view of the nature of geometry. For one can find strong expressions 
of the abstract character of the entities and methods of geometry in the Platonic 
writings. 2 ' The careless habit of assigning feats of insight to such heroes as Plato 
and Pythagoras might easily tempt later writers to overlook issues of simple 
consistency. 

But if we acknowledge, as most scholars now do. that this method cannot be 
attributed to Plato, wc then face the question of who its originator was. Its 
absence from Eratosthenes* report might suggest a later dating. But as wc shall 
later see, it appears to be implied as the background to Diocles* method only a 
short time after Eratosthenes. The Platonic association now assumes new sig¬ 
nificance, for it suggests that Eratosthenes himself played a role in the trans¬ 
mission of this method through its inclusion in a work like the Platonkus. This 
does not lead one to suppose that Eratosthenes invented this device in addition 
to the other mechanical procedure transmitted under his name, but rather that 
he might have introduced it as the centerpiece of a discussion on the nature of 
geometry between Plato and one of his cube-duplicating colleagues. 24 Then which 
colleague? Only Archytas, Mcnacchmus, and Eudoxus arc named in this con¬ 
nection, and of these the first two used methods which would not bring the 
pseudo-Platonic device into mind, as we may gather from Eutocius’ accounts 
of them. This leaves the Eudoxean method, of which wc know only that it 
employed some form of ’’curved lines.** Can one try to identify his solution 
with the pseudonymous one? 

The manipulation of the pseudo-Platonic device, in the manner described by 
Eutocius, does not give rise to any curve. But instead of adjusting the sliding 
beam KL such (hat K is held to the line GB (extended), let us place KL so that 
it will always pass through A (sec Fig. 10(a)]. Then K will trace out a curve, 
and where it meets GB (at E) the configuration for solving the problem is 
obtained. To view its generation a bit more abstractly, wc conceive of two 
parallel rays AK, GH, turning on the given pivots A and G. From the intersection 
H of GH and AB (extended) draw the perpendicular meeting AK in K; then K 
describes our curve. Alternatively, wc may draw from the intersection K of AK 
and GB (extended) the perpendicular meeting GH in H. and H will then describe 
a curve of the same type, differently oriented |see Fig. 10(b)]. In modem works 
these curves arc called ’’ophiurides,” a family of the third order, and the curve 
used by Diocles for solving the cube duplication arises in the limiting case where 
B coincides with either A or G. 23 One may note that a procedure equivalent to 
the pseudo-Platonic method occurs in an Arabic version based, it would appear, 
on an account by the early 2nd-century-A d. geometer Menelaus. 2 * Here, an 
overtly mechanical description in the style of Eutocius, replete with grooved 
beams and pivots, accompanies a parallel description framed in the more abstract 



60 Ancient Tradition of Geometric Problems 


The Geometers in Plato’s Academy 61 




geometric manner just given. Neither actually generates a curve, although the 
method for doing so is evident. The omission might well be due to Mcnelaus, 
for he docs the same thing in his treatment of Archytas’ method. 27 We thus have 
a possible explanation of how a text of Eudoxus* solution ’‘by means of curved 
lines'* might come into Eutocius* hands, now bereft of any mention of curves. 
Moreover, if the text was at all unclear on the positioning of K along line GB, 
one would not see how the discrete proportion GB : BH = HL : LK becomes 
the continuous proportion GB : BD = BD : BK. These two defects marred 
Eutocius' source on the Eudoxean method and so suggest an association with 
the pseudo-Platonic method. 

Under this view of the identity of the two methods, we would have to assign 
to Eudoxus a treatment in the more abstract manner suggested above, for Era¬ 
tosthenes criticizes Eudoxus, no less than Archytas and Mcnacchmus, for having 
addressed the problem in a strictly theoretical manner without thought of practical 
execution. As indicated above, Eratosthenes himself emerges as a likely can¬ 
didate to bear responsibility for casting this method into the form of an actual 




mechanical device. His writing of the PlQtonicus would offer a natural context 
for this and through this association induce the later writers mistakenly to attribute 
the method to Plato. One can detect in Eutocius* version of the pseudo-Platonic 
method certain coincidences of phrasing which link it to other texts he presents. 
Such editorial interventions on his part would be necessary in the event, as we 
propose here, that his source ultimately derived from the relatively informal 
context of a discussion of the Platonic philosophy by Eratosthenes. 

The third of the early solutions mentioned by Eratosthenes entailed the “sec¬ 
tioning off from the cone the triads of Menaechmus." Eutocius presents a text 
of Menaechmus' method in which the desired lines are obtained first from the 
intersection of a hyperbola with either of two parabolas, then, in a second version, 
via the intersection of the two parabolas.” More specifically, if we wish to find 
for two given lines A, B the two lines X, Y which arc the mean proportionals 
between them, then we may construct the parabola satisfying the condition 
X 2 = AY (or, indifferently, the parabola satisfying Y* = BX) and the hyperbola 
satisfying XY = AB. The intersection of these two curves yields the lines X, 
Y which are the desired mean proportionals [see Fig. 11(a)]. Alternatively, we 
may consider the intersection of the two parabolas; for this gives rise to the same 
two lines X, Y [see Fig. 11(b)). 

The essential line of thought, which we have paraphrased here following 
Eutocius, is entirely clear, especially since Eutocius precedes his formal synthesis 
of the problem with an analysis. Assuming the four lines have been set out, 
such that A : X = X : Y = Y : B, if X is set at a right angle to Y, the proportion 
A : X = X : Y associates X and Y as coordinates of a given parabola; similarly 
in the instance of the hyperbola and the second parabola. The notion of setting 
the lines at right angles is quite naturally suggested by the configuration of the 
pseudo-Platonic method which would surely be familiar to Menaechmus, if that 
was indeed the method used by Eudoxus [see Fig. 11(c)]. What has puzzled 
scholars is the manner by which Menaechmus was able to construe these relations 
as properties of sections of cones. Eutocius* text is of limited use in this regard, 
for it is framed entirely in conformity with the Apollonian form of the theory 


62 


Ancient Tradition of Geometric Problems 


The Geometers in Plato's Academy 63 



(c) 

Figure II 

of conic sections. A text adhering closely to the treatment of this problem by a 
4th-ccntury geometer like Menaechmus could hardly employ terminology like 
“latus rectum,” “ordinate,” “asymptote,” and perhaps not even the terms 
“parabola” and “hyperbola.” Thus, modern commentators have engaged in 
ambitious reconstructions of the rudimentary theory of conic curves which Men¬ 
aechmus might have developed for the purposes of his solution of the cube 
duplication.As one can well anticipate, any such theory must have embraced 
a substantial portion of the content, if not of course the form, of Apollonius* 
Books I and 11. 

This effort has provoked surprisingly little uneasiness among historians of 
this period.' 1 Yet apart from this testimonium of the work of Menaechmus, there 
arc but two passages which might reflect the very early study of conics, the one 
in the pseudo-Aristotelian Problems , the other in Euclid’s Phaenomena , and 
hence both datable to around 300 b.c.' 2 At about that same time Euclid and his 
contemporaries were in the position to make a first compilation of the theory of 
conics, of which we possess a good, if indirect, indicator in the works of 
Archimedes." But the tradition of an early theory instituted by Menaechmus 
nearly a half-century before Euclid depends on but a single line of Eratosthenes’ 
verse: “Do not thou seek to section off from the cone (konotomein) the Mcn- 
acchmcan triads.”" Reading this line in its context, we recognize at once that 
the “Menaechmcan triads” (note the plural) refer to the three specific conic 
curves (the hyperbola and two parabolas) associated with each choice of the two 
given lines in the problem of finding the two mean proportionals. Contrary to 
the familiar view, these can hardly refer to the three forms of conic section: 
parabola, hyperbola, and ellipse. For at no known stage of the ancient theory 
were these curves generated together as triplets, while the ellipse has no bearing 
on the ancient solutions of the cube duplication, as transmitted to us, whether 
from Menaechmus or from anyone else. Eratosthenes* term konotomein is no 


guarantee that Menaechmus actually produced these curves as sections of cones, 
for anachronistic applications of terminology have been known to occur among 
historians of mathematics, while Eratosthenes* epigram was not composed as a 
historical effort anyway. Indeed, strictly speaking, Eratosthenes does not even 
claim that Menaechmus sectioned the cone, but only that “you.” the contem¬ 
porary reader of the epigram, need not do so. We thus have good cause to 
consider whether Menaechmus might have constructed his solving curves in 
some other manner. 

An appropriate method is readily conceived on the basis of the pointwisc 
construction of the curves answering to the three relations derived from the 
assumption of the continued proportionality of the lines A, X , K, tf." In the 
operation of the means-finding apparatus attributed to Plato (Fig. 9b), we ma¬ 
neuver the right angle GHZ so that the leg GH always passes through G while 
H always lies on the extension of AB (Fig. 12(a)); if for each position H we 
complete the rectangle HBKN, the vertex N will trace a curve whose coordinates 
are such that BH ( = X) is the mean proportional of GB ( = A)andBK( = Y). 
Similarly, if the right angle AKH is turned so (hat the leg AK always passes 
through A while K lies on the extension of GB (Fig. 12(b)), completing the 
rectangle HBKX, we trace the curve whose coordinates arc such that BK is mean 
proportional of HB and BA ( = B). Thus, where these two curves intersect, the 
coordinates X, Y will be the two mean proportionals between the given lines A, 
£, as required by the problem. In an alternative construction, we can figure Y 
as the half-chord perpendicular to the diameter of a circle such that Y divides 
that diameter into segments of length B and X (cf. Elements VI, 13 or If, 14 
(see Fig. 12(c)). To produce the curve associated with the relation K 2 = BX , 
then, one sets out as many lengths for X as one wishes, determines Y for each 
in the above manner, locates a point for each such pair of coordinates, and 
connects the points by means of a flexible ruler or other suitable drawing aid. 
The construction of the curve associated with the relation X 2 = A Y is of course 
the same |$ec Fig. 12(d)]. In the case of the relation XY = AB, one may find 
Y coordinate to X via the “parabolic application of area,” based on the fact that 
the diagonal of the rectangle of sides B + X and A + Y passes through the 
origin (cf. Elements I, 44) (see Fig. 12(e)), Otherwise, the generation of a curve 
through a selection of values for X follows the same pattern. Having drawn two 
of these curves over the given lines A, B, one determines from their point of 
intersection the lines X, Y which solve the problem of the mean proportionals. 

Pointwisc procedures of this sort are familiar among later writers. Diocles, 
for instance, constructs his own curve for solving the cube duplication by de¬ 
termining a set of points and then connecting them via a flexible ruler; his 
alternative construction of the parabola, framed in effect around its focus-directrix 
property, is also in the pointwise manner. 36 Eutocius describes analogous pro¬ 
cedures for drawing all three kinds of conics, and notes that writers on mechanics 
often adopt such methods for the conics where other methods do not work well.*' 7 
Similarly, Anthemius marks out the contours of burning mirrors by orienting 
the tangent lines at selected points. 38 No source identifies the originator of these 



i 


64 Ancient Tradition of Geometric Problems 



pointwise methods. But Diodes* dependence on Menaechmus is evident in his 
presentation of a form of the two-parabola solution for the cube duplication, and 
one can readily assume that he also found with Menaechmus a precedent for his 
use of a pointwise construction of those curves. 39 It is right, I believe, to view 
this method of constructing curves to be an insight of significance in the devel¬ 
opment of geometry, yet hardly overreaching the abilities of prc-Euclidean geom- 


Thc Geometers in Plato's Academy 65 



eters. Indeed, its invention prior to Menaechmus, or perhaps by him, is rendered 
plausible by the association with Diodes. 

I propose, then, that Menaechmus based his solution on curves defined with 
respect to the second-order relations among the mean proportional lines. This 
does away with the warrant to speculate on how Menaechmus might have formed 
these curves as sections of cones and which of their properties he might have 






Ancient Tradition of Geometric Problems 


Roved, exercises which have all too long distracted historians from considering 
evidence from around the time of Euclid, a more plausible context for the first 
ttudy of these sections. This view has the welcome corollary of restoring con- 
Idence in the reliability of Eutocius, our principal source of documentary ma- 
trials on this phase of early geometry. Although his account of Menaechmus 
^questionably reflects the reworking effort of a post-Apollonian editor, it serves 
tow as an accurate guide to the essential line of thought Menaechmus followed. 


| geometric methods in the analysis of 

I PROBLEMS 

Jfhe set of 4th-century solutions to the cube duplication already exploit a wide 
tiamge of geometric methods which would remain characteristic features of the 
indent problem-solving tradition. Each of the methods considered, for instance, 
introduces one or more special curves, generated either through the section of 
^plids, as in Archytas’ solution, or through the mechanical motion of line seg¬ 
ments, as in the pseudo-Platonic arrangement, or through pointwise construc¬ 
tions, as proposed for the solution by Menaechmus. Eutocius* text of Menaechmus* 
yolution highlights two further aspects of methodology not displayed in any of 
is other ten texts. One is its use of the second-order relations of lines associated 
ith the pre-Euclidean technique of “application of areas,*’ on which was built 
he later theory of the conic sections. The second is its adoption of the two-part 
Ormat of “analysis and synthesis*’ which would become, in the hands of Apol- 
nius and his contemporaries, a powerful instrument for the discovery and 
reposition of the solutions to problems. 

The “application of areas,” as wc learn from Proclus on the authority of 
udemus, owes its origins to the ancient Pythagoreans, although Plutarch knows 
i f a tradition assigning its invention to Pythagoras himself, 4,1 It relates to a series 
of problems now extant in Euclid’s Book VI and supporting lemmas to be found 
in his Books I and II. The objective is to “apply” (parahalfoin) a given area to 
P given line segment (1,44) or, alternatively, «o construct a parallelogram similar 
lo a given figure and equal to a second given figure (VI, 25). One may also seek 
P apply the given area to a given line segment such that the resulting parallel¬ 
ogram either ‘‘exceeds” ibyperbolfoin) or “falls short” (eUeipein) by a figure 
nmitar to a given parallelogram (VI, 29, 28, respectively; cf. II, 6, 5). In the 
sarlier stages of this technique, one would take the figures of application to he 
tctanglcs and the figures of excess or defect to be squares. Thus, for the first, 
>r “parabolic** case, one seeks to construct tltc line X such that AX = M for 
» given line segment A and a given rectilinear area M, c.g., the square of side 
I. In the “hyperbolic” case, one seeks X such that X(X f A) - B*. and in 
“elliptic” case, the X such that X(A ~ X) = B 2 |sce Figs. 13(a), (b)|. 4 * 
Tie attribution to the Pythagoreans merely confirms what one infers from Euclid’s 
foments: that these techniques were familiar in the pre-Eudidcan geometry, 
iat is, in the 4th century n r and perhaps the latter part of the 5th century tic. 
ul 11 * s °ow well known that the same problems, treated in the form of numerical 



Figure 13 

solving procedures, were a prominent part of the Babylonian tradition more than 
a millennium earlier. 12 One would thus infer that the Greeks first learned these 
techniques through contact, direct or indirect, with the Babylonian tradition. 4 ' 
The same would also have provided them a precedent for working out ordered 
procedures for the solution of problems ; but the specifically geometric form into 
which the Greeks cast their own version of this technique is not manifest in the 
Babylonian texts now extant. 

A most notable example of the “application” technique is in the division of 
a line into “extreme and mean ratio” (VI, 30; cf. II, 11), through which one 
can effect the inscription of a regular pentagon in a given circle (IV, 10, II) 
and the inscription of the regular icosahedron and dodecahedron in a given sphere 
(XIII, 16. 17). In all these instances, Euclid’s treatments are in the synthetic 
manner. But Eutocius* text of Menaechmus* solution of the cube-duplication 
suggests through its adoption of the double trunk that these other problems might 
also have owed their initial solutions to analytic treatments. Indeed. Pappus 
preserves a set of alternative constructions of the inscribed regular solids effected 
by means of analysis and synthesis. 44 Proclus relates that Plato taught the method 
of analysis to Ix'odamas lor use in his geometric researches and that Eudoxus 
used analyses in his studies of the “section” (tomi). A> 

It will be useful to reconstruct an analysis along the lines suggested by Euclid’s 
synthesis of the problem of inscribing the regular pentagon in a given circle. As 
wc shall later see, a comparable analysis is discernible as the basis of Archimedes* 
solution of the inscription of the regular heptagon (Chapter 5). Assume then that 
the regular pentagon ABCDE has been inscribed in the circle |sec Fig. I3(c)| 
and draw the diagonals Al), BE meeting in F. In the triangle AFK the angles 
at A and E arc equal, intercepting the equal arcs DE, AB, respectively. Since 
angle BAD intercepts the double arc BCD, it will be double the angle FAR. 
Since, furthermore, the angle BFA is external to triangle FAE, it will equal the 
sum of angles FAE, AEF, that is, twice angle FAE. Thus, angles BAF, BFA 
arc equal, so that AB = BF. Since F is the vertex of the isosceles triangle FAE, 
it lies on FG. the perpendicular bisector of the given side AE. Its position may 
thus be given via neusis , in which the intercept FB from a line drawn from the 




given point E shall have a given length (= AB) and lie between a given line 
(FG) and a given circle (that with center A and radius AB). The corresponding 
synthesis is evident, and in fact is nothing other than a variant of Hippocrates' 
third lunule construction. 46 

Euclid's method in IV, 11 does not employ a ueuxis. however, so that the 
implied analysis must have followed a different course. If we adopt the same 
figure as before, we have that the triangles ABE, FAE arc similar, since their 
base angles (at B, E, and A) intercept the equal arcs AE, AB, ED, respectively. 
Thus, AB : BE = EF : AE, and since AB = AE = BF, it follows that BF 2 
« EF* EB. We have thus reduced the problem of inscription to one of dividing 
a line into segments satisfying this second-order relation. The latter problem, 
familiar to the ancients as the division into "extreme anil mean ratio” (cf. 
Elements VI, Dcf. 3), and to us as the "golden section," can he effected via 
the methods of the "application of areas," as Euclid docs in II, II and again 
in VI, 30. 47 In the latter, the given area BE 2 is applied hypcrbolically to the 
given line BE; that is, BE 2 = X(BE + X) (sec Fig. 13(d)|. Setting BF = X, 
t follows that BF 2 = BE* EF, as required. But in II, II an independent method 
s used where BF is constructed as the difference of x /i BE and the hypotenuse 
tf the right triangle of legs BE, Vi BE (see Fig. 13(c)|. 4 " The alternative method 
vould seem to indicate an origin at an earlier stage of the development of these 
cchniques. 49 

If Euclid's procedure drew from the neusis -related analysis just given, wc 
rould expect a synthesis like this: divide the given line BE at F as specified in 
ic analysis; complete the isosceles triangle with BE as base and legs BA, AE 
ach equal to BF; circumscribe the circle about BAE; then BA. BE are the side 
ad diagonal, respectively, of the regular pentagon inscribed in this circle. If 
e then inscribe in the given circle an angle equal to ABE, wc shall have 
rtermined an arc A'E' subtended by a side of the regular pentagon inscribed 


I 



in it, fmm which the figure is readily completed. This approach is of course 
entirely within the scope of the early techniques; it is sufficiently straightforward 
in conception to make one uncomfortable in denying that the ancients knew it. 
But the synthesis given by Euclid in IV, 11 is quite different. We thus receive 
no encouragement from him in supposing the ancients followed this method, 511 
and must frame another analysis as the basis of his method. 

Assume again the regular pentagon ABCDE inscribed in the circle and now 
draw the diagonals AC. CE |scc Fig. I3(f)|. It is evident that the triangle ACE 
is isosceles and that each of the angles at its base is twice the angle at its vertex. 
The inscription thus reduces to the construction of an isosceles triangle of this 
form; the synthesis ol the latter problem is given by Euclid in IV, 10 in order 
to effect the inscription in IV, It. To construct the triangle, we can produce the 
following analysis: assume ACE is isosceles, where the angle at A is twice that 
at C |sec Fig. I3(g)|. I .cl AFbisect the angle at A, whence triangles AFE, CAE 
are similar. *!1uis, EF : EA = EA : AC, or HA 2 = EF* AC. Now, EA = FC, 


C 



C 



<B> 


i 

i 







70 Ant'tcm Tradition of Geometric Problems 



(h) 


Figure 13 


each being equal to line AF in the isosceles triangles AFE, FAC, respectively. 
Furthermore, AC = EC. Thus, FC 2 = FE • EC. In this way, the construction 
of ACE reduces to the division of a given line CE in accordance with the extreme 
and mean ratio, as given in II, 11 (or VI, 30). 

I have no doubt that the analysis underlying IV, 10—by which the construction 
of CAE is reduced to finding F such that FC 2 = FE • EC—took this form. The 
techniques of proportions and similar figures were already prominent in the 
ancient geometry quite early, as we have seen through the problem-solving efforts 
of Hippocrates. Moreover, treatments survive of closely related problems which 
adopt precisely this configuration of similar triangles. 51 Nevertheless, the syn¬ 
thesis Euclid presents in IV, 10 docs not merely retrace the steps of this analysis. 
Euclid assumes the division of CE such that CF 2 = FE • EC and completes the 
triangle CEA such that EC = CA and HA = FC |scc Fig. 13(h)); it is to be 
shown that the angles at A and E arc each twice that at C. Draw AF and introduce 
the circle AFC. Since EA 2 = FE • EC, EA is tangent to the circle [Elements 
HI, 37), whence angles FAE and FCA are equal (III, 32). Euclid next proves 
that angles EFA and EAC are equal; 52 since the triangle ECA is isosceles, the 
latter angle equals FEA, so that the triangle FEA is also isosceles, whence EA 
= AF. But by construction, EA = CF; angles FCA and FAC arc thus equal 
base angles in the isosceles triangle FAC. The exterior angle EFA equals their 
sum, or twice the angle at C. Thus, in the triangle CEA, the base angle EAC 
(which equals angle EFA) is twice the angle at the vertex C. 

Our reconstruction of the analysis has not prepared us for Euclid's introduction 
of the circle and the angles formed by its tangent. But these moves are quite 
naturally motivated by Euclid’s pedagogical decision to defer the presentation 
of the techniques of proportion until later in his treatment of plane geometry 
(i.e. t in Books V and VI). The constructions of the regular polygons in Book 
IV must thus be effected exclusively via methods of congruence. Thus, the 


1 nc 1 winiicit'iN m i i.mt > /h.iiU'iii) i i 

equality of angles EAF and ACE, which could have followed at once from the 
similarity of the corresponding triangles through the proportionality of their sides, 
must now be secured via the tangent EA to the circle AFC. This discrepancy 
raises a subtle point for the interpreter. Euclid's formalized organization of the 
Elements need not have followed the order of discovery; although it is conceivable 
that the whole body of plane geometry in Books 1—IV was worked out prior to 
the adoption of proportion techniques by the ancient geometers, I find that only 
remotely plausible as a position on the historical development of the field. Since 
the purpose of an analysis of a problem is fundamentally heuristic, it is entirely 
feasible that this part utilize methods formally excluded at the stage of synthesis. 
For it is the synthesis alone which has formal status. In the case of the construction 
in IV, 10 at issue here, one could fashion an analysis which conforms step for 
step with Euclid's synthesis. 51 But that would be an exercise without cither 
pedagogical or heuristic interest. In the light of this, Euclid's omission of analyses 
in his treatments of construction problems in the Elements follows naturally from 
the editorial decision to formalize the exposition of this material. 

To the extent that the Elements do reflect the earlier procedures—certainly 
in more general respects, if not always on points of specific detail—we may 
submit the reconstructions above as a specimen of the problem-solving field 
among the 4th-century geometers. Characteristic of the method arc the exploi¬ 
tation of analyses and the reduction of the desired configurations to conditions 
relating to the application of areas. From this period wc may cite two further 
witnesses to the adoption of this approach: one is a passage from Plato's Mena , 
the other a set of remarks by Proclus on a geometer named Amphinomus. 

The Meno (86 c - 87 b) introduces a method of reasoning 4 * from hypothesis" 
which, but for its name, is identical to that of “reduction" as used, for instance, 
by Hippocrates in his attack on the cube duplication. The issue in the dialogue 
is it) establish whether virtue is teachable. In the absence of a satisfactory def¬ 
inition of what virtue is, Socrates proposes that one consider a hypothesis from 
which to pursue the examination of the main question. 

t say “from hypothesis” in the manner that the geometers often make inquiry, 
whenever someone has asked them, for instance about an area, whether this area 
here can be stretched out as a triangle in this circle here, one would say ‘I don't 
yet know whether this is of such a sort, but t think that as a certain hypothesis the 
following will assist in the matter. If this area is such that the one who has stretched 
(it) along its given line (makes it) fall short by an area such as is the stretched 
(area) itself, then it seems to me that a certain result follows, but a different result, 
if it is impossible that these things be done. Having hypothesized, then, I wish to 
say to you whether the result about its stretching in the circle is impossible or 
not." 4 

Like all too many of the mathematical passages in the dialogues, this one leaves 
much to be desired in terms of clarity. Plato’s phrasing here is quite free, whether 
by reason of his deliberate choice to suit the colloquial context of the dialogue, 
or perhaps through imprecision in the mathematical terminology of his day. For 
instance, the procedure of “stretching m“ [entcinein) is clearly the inscription 





72 Ancient Tradition of Geometric Problems 



Figure 14 

of one figure within another. Hut can one assume that the procedure of “stretching 
along *’ (paruteinein) refers to the application [paraballein) ol area, as would 
appear to be the case here? Is the area which falls short (elteipein) “by such as 
is“ (toioutoi, hoion ei) the applied area similar to it, or equal to it? Does “its 
given line** signify the diameter of the given circle, or perhaps some other line 
serving as a side for the figure of application? 

These obscurities have occasioned dozens of efforts of interpretation. Hut one 
of the oldest attempts still remains the most plausible and widely accepted.'' 
The problem as initially posed requires the inscription of a given area as a triangle 
in a given circle. We may figure the given area as a square ol side B anil denote 
by A the diameter of the given circle |scc Fig. I4|. 1*hcre is no loss of generality 
in requiring that the inscribed figure lake the form of an isosceles triangle. In 
accordance with the mode of analysis, assume the figure has been inscribed as 
specified; let the altitude X of the triangle lie along the diameter, and let V be 
half its base, set at right angles to the diameter. Hie condition on the area thus 
stipulates that XY = B J . Furthermore, since Y is a half-chord in the circle, Y 
is the mean proportional between X and A - X, the two segments of the diameter 
it cuts off; that is. X : Y = Y : A - X. In this way the problem is reduced to 
another one, corresponding to that stated in the second part of Plato's passage: 
the given area B 2 is now to be applied to the given line A such that the area by 
which it falls short (namely, the rectangle of sides Y and A - X) be similar to 
the area applied (the rectangle of sides X and Y). Thus, the passage may be 
viewed as based on this reduction or analysis of the problem of inscription. 

If we complete the analysis, in the manner suggested by Menaechmus’ treat¬ 
ment of the cube duplication, we will observe that the condition on areas (XY 
» B 2 ) entails that X and Y arc coordinates in a given curve, namely, the 


The Geometers in Plato's Academy 73 


hyperbola having as its asymptotes the diameter of the circle and the tangent 
drawn at its endpoint, where the hyperbola passes through the point whose 
coordinates both equal B. We have proposed that Menaechmus introduced such 
curves in the form of pointwisc constructions corresponding to the defining area 
relation. But whatever construction lie used, it would perfectly suit the purposes 
of the problem discussed mi the Meno. This effectively derails the principal 
objection raised by some against this interpretation of the passage: that it supposes 
a problem incapable of solution by (lie methods available to prc-liuclidcau 
geometers.'* 

A remarkable feature of the Meno passage is that it expresses the mathematical 
project not as the actual solution of a problem, but rather as the determination 
of the possibility of its solution. This has led many to view the passage as 
discussing a “diorism.” that is, the statement of the necessary condition for the 
solvability of a ptohlcm. But in the mathematical literature diorisms have the 
form of explicit conditions on the givens of the problem. In the present ease, 
this might be the statement that the given area must be less than the area of an 
equilateral triangle inscribed in the given circle; having verified this relation to 
hold for particular values of the givens, one would know that the problem is 
solvable in this ease, even before one hits begun the solution of (he problem as 
such. Although it is often the ease that the analysis of a problem reveals the 
appropriate form of its diorism, nevertheless, the articulation of the diorism is 
quite different from the analysis or reduction of the corresponding problem. We 
thus have to explain why Plato here frames this example of problem reduction 
as if it were equivalent to the determination of possibility. 

According to the usual chronology of Plato’s works, the Meno is transitional 
between the earlier and the middle dialogues, so that its composition is placed 
around 385 nr %t This is two decades or more before the activity of Menaechmus, 
a disciple of Kudoxus whose association with the Academy falls in the period 
after 370 fix*. Some of the geometric techniques available to Menaechmus would 
not yet have been introduced at the earlier time of the Meno, and these might 
include the methods of curve drawing discussed above. T hus, Plato’s emphasis 
on the possibility of the inscription problem might be taken to signify that 
geometers had then discovered the diorism, hut not the actual solution of this 
problem. For a proof of the diorism can be effected via elementary methods of 
construction,'* but this is not true of the inscription problem itself. With the 
introduction of the methods affiliated with Menaechmus’ cube duplication, not 
only could the problem now he solved, but one also would obtain an alternative 
proof of the diorism; for (he problem of application is possible only when the 
auxiliaiy hyperbola (in (lie derived problem) intersects the circle, and in (he 
limiting case, where the curves meet at a single point of mutual tangcncy, one 
finds that X = % A, equivalent to the ease of the equilateral triangle. w 

Supporting this view is one’s intuition that the successful solution of the 
inscription problem could not long have antedated Menaechmus’ solution of the 
cube duplication. But (he form of the Meno passage must surely have been 
affected by certain dramatic and dialectical considerations at least as much as 



74 Ancient TimJition of Geometric Problems 


by these historical and technical ones. Plato's principal concern in the dialogue 
is over a question of possibility: the teachability of virtue. It thus suits his purposes 
to frame the geometric example in like manner, as a determination of the pos¬ 
sibility of a construction, even though the geometric method illustrated is that 
of reduction for the solution of problems. Further, Plato's account here is likely 
to reflect his own interpretation of what the true goal of such geometric activities 
is: not the finding of constructions, but rather the establishment of (heir possi¬ 
bility, that is, the demonstration of the existence of the solving entities. In the 
later mathematical writings problems arc often asserted in the form, "it is possible 
(dynaton estin) to construct...," and taken literally, this might indicate an 
existential motivation underlying the activity of problem solving. I argue below 
(cf. also Ch. 8) (hat this provides a poor account of the objectives of the geometric 
research in this period. But geometers' actual motives need not much have 
affected Plato's view of them. I think that the special concerns of his general 
philosophy could well have encouraged Plato to represent, or misrepresent, the 
geometers' search for constructions as essentially an investigation into (heir 
possibility. This would account for the discrepancy between the Mena passage 
and the standard format in mathematical writings. 

Proclus reports that Leon, a younger contemporary of Plato, "found dior- 
isms," so that wc may infer additional evidence of interest in such studies of 
problems (in Euclidem, pp. 66 0. Proclus elsewhere provides an example further 
revealing the use of techniques of area application in the analysis of problems. Wl 
A certain geometer named Amphinomus is said to have distinguished problems 
as "ordered," "intermediate," or "unordered" according to whether they permit 
of one, several, or an unlimited number of solutions. By way of illustrating this 
last kind, Proclus cites the problem of dividing a given line segment into three 
parts in continued proportion. If one divides (he given line into segments in the 
ratio of 2 : I and one applies the square of the smaller segment to the larger 
segment as to fall short by a square, then the given line will be divided into 
three equal parts which arc thus in continued proportion. If the given line is then 
divided into segments in any ratio greater than the double, say the triple or any 
larger multiple, and if in similar fashion the square on the smaller segment is 
applied to the larger as to fall short by a square, then again the line will have 
been divided into three parts in continued proportion. Let us designate by X the 
larger of the initial segments, by Y the smaller; then the procedure of application 
produces Z such that Y 2 * Z(X - Z). It thus follows that the segments Z, Y, 
X - Z are in continued proportion, while their sum equals the given line. Since 
the application is possible only when Z is less than or equal to one-half X, 
whence the maximal value for Y is seen to be one-half X, Proclus’ initial choice 
of the division X : Y » 2 : I represents the limiting condition for the possibility 
of solving the stated problem. One would suppose that a "diorism” to this effect 
had been worked out in the original treatment. But as the number of larger 
multiples is unlimited, each yielding a solution, the problem has an infinity of 
solutions and so falls under Amphinomus* class of "irregular" (or "unordered," 
atakta) problems.*' 


I tic Geometers iti Plato's Academy 7S 


This is a nice specimen of the use of application techniques for problem¬ 
solving. A 4th-century provenance is indicated, but not entirely secured. One 
would suppose that Amphinomus himself cited this problem as an instance of 
his "unordered" class, although its appearance in this context might possibly 
be due to a later editor like Proclus. What we know of Amphinomus otherwise 
is contained in three additional references by Proclus, all of which associate him 
with 4th-ccnlory figures. Amphinomus is said to have upheld the view of Ar¬ 
istotle, that geometry docs not inquire into the causes of things (in Eucl .. p. 
202). 62 Me is linked with Spcusippus, Plato’s nephew and successor, as main¬ 
taining that all propositions in geometry should be framed as theorems, for this 
form best captures the eternal and nongencrable nature of geometric entities; by 
contrast, Proclus continues, the mathematicians allied with Mcnaechmus em¬ 
phasized the problematic element in geometric research, whether this seeks the 
production of an entity or the determination of its properties (in Eucl pp. 77 
f).*' Furthermore, when Proclus observes that not every proposition has a true 
converse, hut that a proposition will be convertible when it refers to the primary 
or essential attribute of a class, he adds that "these things did not escape the 
notice of the mathematicians around Mcnaechmus and Amphinomus" (In Eucl.. 
p. 254). This passage recalls a discussion by Aristotle, to the effect that a proper 
definition should pick out attributes which apply to all and only the members of 
the class defined (Posterior Analytics I, 4, 5). The special interest which geom¬ 
eters might have in the convertibility of propositions is clear from another com¬ 
ment by Aristotle: 

If it were possible to prove something true from something false, the procedure 
ol’ analysis ianatynn) would he easy; for it would convert of necessity. For let us 
posit that A obtains; from this supposition these things follow which t know do 
obtain, namely It. Then front these things I can show that the funner (At obtains. 
Now the iliings in mathematics tend to he convertible, for they posit nothing 
contingent, but only definitions—and in this they differ from the things in dialectical 
philosophy Ubitl.. I, I2).* 1 

Aristotle here portrays analysis as a deductive procedure leading from the desired 
premise (or construction), hypothesized to be true (or constructible), to another 
one known to be true (or construeiiblc). This accords with the examples we have 
already discussed, as well as with those in the later mathematical literature. 
Moreover, Aristotle’s awareness of the relevance of the issue of convertibility 
reveals that current mathematical practice had recognized the need for the cor¬ 
responding synthesis to effect the demonstration; where steps in the analysis are 
not simply convertible, one would have to idcniify the conditions under which 
conversion is valid, that is, the conditions of the diorism. Aristotle thus indirectly 
attests to the maturity of this method among the geometers of his generation. 1 * 
Wc thus sec that Mcnaechmus* method foi solving the cube duplication 
typifies an important field of the prc-Euclidean geometry. It is but one of several 
known instances employing the technique of application of areas, a versatile 
instrument lor the analysis of problems. Indeed, the field of analysis had been 




^ jTt .*.^*.~. «.***, ^ftVc*ft» { l AS 7 W " 


76 Ancient Tradition of Geometric Problems 


The Geometers in Plato's Academy 77 


sufficiently advanced by around the middle of the 4th century B.c. that certain 
special features of the logical structure of analysis began to attract the attention 
of philosophers. For instance, the “diorism" expressing the conditions for the 
possibility of solution results from consideration of the conditions permitting the 
steps in the analysis to be converted, so to produce the formal synthesis. Among 
some there already appears to lie emerging a self-conscious adherence to the 
formalist ideal. In the question as to whether one ought to frame geometric 
propositions as theorems or as problems, Proclus finds a spokesman for the 
theorematic view, as one might predict, in the person of a 4th-ccniury Platonist 
philosopher. But he draws from the geometer Menacchmus a sense of the priority 
of problems. This division is surely ominous. For a preoccupation with the 
formal exposition of theorems, at the expense of the analysis of problems, could 
hardly avoid retarding the effort to discover new solutions. On the other hand, 
the formal work of synthesis can sometimes raise difficulties demanding geo¬ 
metric insight for their resolution. The efforts of Eudoxus, to which we now 
turn, illustrate this impressively. 

EFFORTS TOWARD THE QUADRATURE OF THE CIRCLE 

As geometers sought ways actually to construct (he square equal to a given circle, 
they came to recognize that the underlying theorems on circle measurement 
posed difficulties of their own. Hippocrates* measurement of the lunulcs, for 
instance, must assume that circles have the ratio of the squares on their diameters. 
Any proof which Hippocrates could have offered for this theorem, such as the 
one we proposed above on the model of Antiphon's sophism on the circle 
quadrature, must inevitably have adopted a naive manner of limits, for the 
formally correct method, as presented in Euclid's Hook XII, owed its first in¬ 
troduction to Eudoxus. 

Aristotle names the 4lh-ccnttiry Sophist Bryson, along with Hippocrates and 
jj Antiphon, as having attacked the problem of circle quadrature.'*’ But the later 

; commentators Alexander (3rd century a.ix) and Proclus. Ammonius, and Phil- 

oponus (5th and early 6th centuries), who inform us of the details of Bryson’s 
argument, seem to know only this much*': having circumscribed a square about 
the circle and (hen inscribed another within it. and having taken a third square 
intermediate between the first Iwo, Bryson claimed that this square equalled the 
circle, “for things greater than and less than the same things arc equal." 1 * On 
the meaning and criticism of this argument, the commentators disagree. The 
simplistic reading leads at once to absurdity: one might claim on the same basis 
that 9 and 10 are equal, since they are both greater than K and less than I2." 1 
A more sophisticated interpretation takes into consideration the sequence of all 
circumscribing and inscribed regular polygons: then “the rectilinear figure” 
which is less than all the circumscribing polygons and greater than all the in¬ 
scribed polygons will equal the circle, since the circle satisfies the same inequalities. 

Philoponus dismissed this latter effort by Proclus to refurbish Bryson’s prin¬ 
ciple in the form “to that than which there is a greater and a lesser there is also 


an equal.“ m It is hard to see the force of Philoponus’ attempted counterexample, 
however. To be sure, the “hum angle” contained between a circle and its tangent 
is less than every rectilinear acute angle, while the corresponding mixed angle 
between the circle and one of its diameters is greater than every rectilinear acute 
angle. But when he observes that this has not established the existence of an 
equal angle/' we might well ask what angle he is referring to. Through further 
elaboration of this class of mixed angles, Philoponus shows that he would deny 
what the modem theory of curves would affirm: that the angle at which two 
curves intersect each other is taken equal to the angle between their tangents at 
the point of intersection. Despite the fact that these angles arc obviously different 
from each other geometrically, it is important to accept them as quantitatively 
equal if quantitative statements ahoul tangents arc to be possible at all. It is 
interesting that writers in the ancient tradition of geometric optics adopt this 
principle quite freely, manifesting none of the qualms expressed here by 
Philoponus.” 

Philoponus further objects to Proclus’ interpretation of Bryson on the grounds 
that it misses the point of the problem of circle quadrature: 

Those squaring the circle did not seek (to establish) whether it is possible that there 

exists a square equal to the circle, but rather Blinking that it can he so, they tried 

to produce a square equal to the circle ,l 

But in accepting Proclus* argument as an existence proof, he is perhaps too kind. 
For Proclus has far too easily slipped in the “rectilinear figure” presumed at 
once greater than all the inscribed figures and less than all the circumscribed 
ones. If. for instance, this intermediate figure is being generated along with the 
sequence of bounding polygons, then it will effectively turn out to be an infinite- 
sided regular polygon, namely the circle itself. But a subtle modification of the 
argument can avoid this difficulty. For each bounding polygon, let us construct 
the equal square. As the number of sides increases, the sequence of inscribed 
polygons gives rise to a sequence of squares of increasing area, while the se¬ 
quence of circumscribing polygons gives rise to a sequence of squares of de¬ 
creasing area. Since, furthermore, all the squares in the one sequence arc less 
than all the squares in the other, there will be a square intermediate between the 
two sequences. That the difference between the circumscribed and the inscribed 
polygons can be made arbitrarily small is geometrically obvious and can be 
secured by a relatively straightforward bisection argument/ 1 lienee, the inter¬ 
mediate square is unique, from which its equality with the circle follows from 
Bryson's principle. In this way, a Weierstrassian procedure lor defining the 
magnitude of the circle might be foreshadowed, if dimly, in Bryson’s approach. 
This is the interpretation adopted by some modem writers/ 5 But we do well to 
note that the ancient commentators on Biyson seem to have no linn account of 
the details of his argument and do not themselves give a precise statement of 
his principle and a valid fonn of its application, despite their own acquaintance 
with the limiting methods in the tradition after Eudoxus. Indeed, the issue at the 
heart of Bryson’s sophism, the articulation of the conditions for equality between 




78 Ancient Tradition of Geometric Problems 


A 

P 


(b) 


Figure 15 




continuous magnitudes, receives no fully satisfactory explication cither among 
the philosophers or among the geometers of antiquity. One encounters instance 
upon instance in the writings of Euclid, Archimedes, and others where notions 
of continuity enter, based on intuitive assumptions rather than on explicit 
formulations. 7h 

Bryson's argument must reflect an interest among mid-4th-ccntury geometers 
in the formal questions entailed by the assumption that problems like the circle 
quadrature do indeed have solutions. In this context Eudoxus devised a special 
technique, called by modem writers the “method of exhaustion" or the “indirect 
method of limits.“ which could secure the demonstration of theorems like those 
required by Hippocrates, even if it left in abeyance some of the deeper foun¬ 
dational issues implicit in Bryson’s argument. The Eudoxean method, as we 
learn principally from Archimedes, is the basis of the measurements of the circle, 
pyramid, cone, and sphere in the manner now extant in Euclid’s Book XII. 
Archimedes himself, whom we may perceive to be working from Eudoxus’ 
treatments in preparing such early works of his own as the Dimension of the 
Circle , applies essentially the same technique one finds in the Elements . 77 

Let us then sketch Euclid’s theorem on the area of the circle (Elements XII, 
2) as a representative instance of this method. One proposes to demonstrate that 
circles have the ratio of the squares on their diameters. Let the circles be A, B, 
and the squares be C, D |sec Fig. 151- If not truc A : B = C : D, then 
for X such that A : X = C : D, X will either be less than or greater than B. 
Assume first that X is less than B. We now inscribe in B a regular polygon, 
form from this the inscribed regular polygon having twice as many sides, and 
continually repeat this procedure on the newly formed polygons. Since the dif¬ 
ference between the circle and the polygon is reduced by more than half at each 
step in this procedure, 78 we will eventually obtain a polygon Q such that B - Q 
is less than B - X 79 ; that is, Q is greater than X. Now inscribe in A the regular 


The Geometers in Plato's Academy 79 


polygon P similar to Q. Since P : Q = C : D = A : X and P is less than A. it 
follows that Q is less than X. This contradicts our having found Q greater than 
X. Hence, X cannot be less than B. One might conceive an exactly analogous 
procedure utilizing circumscribed polygons whereby the alternative assumption 
that X is greater than B is reduced to a contradiction. But Euclid perceives a 
much simpler alternative. Using the same notation as before, we posit the quantity 

Y such that Y:B = A:X = C:D. Since X is greater than B, Y is less than 
A, so that wc are in precisely the same situation as before: the existence of such 

Y leads to a contradiction, so that X cannot be greater than B either. Hence X 
equals B, and the stated proportion of the circles and the squares of the diameters 
is thus affirmed. 80 

The procedure of inscribing regular polygons and successively doubling the 
number of sides has an evident precedent in Antiphon's circle quadrature, al¬ 
though Eudoxus surely drew it from Hippocrates* proof of the theorem on the 
ratios of circles, 81 for Eudoxus’ proof had the effect of rigorizing the treatment 
of this very same theorem. The key insight in the revised version is the appeal 
to successive bisection in the context of an indirect proof. The lemma that any 
finite magnitude can be reduced by repeated bisection so as to become less than 
any assigned finite magnitude of the same kind is characteristic of all the limiting 
arguments in Elements XII and is retained in the majority of limiting theorems 
of Archimedes. It is proved in Elements X. 1; but the method employed there 
docs not conform to the style in Book XII.One detects that this lemma could 
be deemed sufficiently obvious as not to require an explicit proof at the initial 
stages of Eudoxus' theory, but that later disciples sought to ground it on a more 
fundamental axiom, comparable to the “Archimedean axiom” on continuous 
magnitudes. 

It is worth noting certain features which do not characterize Eudoxus’ method. 
Convergence is always one-sided, usually via a sequence of inscribed figures. 
The device of two-sided convergence, via sequences of paired inscribed and 
circumscribed figures approaching each other arbitrarily closely and hence ap¬ 
proaching a common limit, is typical of Archimedes’ more advanced measure¬ 
ments, but appears to have no precedent in earlier technical writings. 8 ' It thus 
seems implausible that Bryson’s sophism on circle quadrature included a clear 
representation of the two-sided manner of limits, so that the relatively unso¬ 
phisticated form of the argument suggested by the report of Alexander (where 
the inscribed rectilinear figure continuously increases toward the circumscribed 
figure, and so at some time passes through a figure equal to the circle in mag¬ 
nitude) seems more likely to capture Bryson’s attempt than the more elaborate 
version proposed by Proclus. Another striking feature of Eudoxus’ method is its 
failure to utilize the forms of the theory of proportion given in Elements V. For 
comparison, wc may reconstruct an alternative treatment in the Euclidean manner 84 : 

To prove A : B = C : D. let us suppose the contrary. Then, first, let A ; B be 

greater than C : D. There exist integers m, n such that mA > nB. while mC £ 

nD.* s Since A > n/mB, we may inscribe in A a regular polygon P such that P > 

n/mB.“ Let Q be the polygon similar to P inscribed in B. It follows that mP > 





80 Ancient Tradition of Geometric Problems 


The Geometers in Plato’s Academy 81 


wQ. Now P and Q arc similar, so that P : Q = C : D; since mC ^ nD, one has 

mP < iiQ. This contradicts the previous inequality relating P and Q. If. then. A 

: B is less than C : D. the inverted ratios satisfy the opposite inequality, so that 

the same contradiction results. Thus, A : B = C : D. 

The most straightforward reason for Euclid's failure to adopt such a procedure 
in (he ease of this and comparable theorems in Book XII is that Book XII owed 
its origin to earlier work than that which yielded the proportion theory presented 
in Book V. While the latter is usually ascribed to Eudoxus, an alternative pro¬ 
portion theory conforming quite closely to the style adopted in Book XII is 
implicit in certain proofs In Archimedes and other later writers. 117 One would 
thus infer that Book V reflects modifications in Eudoxus' theory by his followers, 
but that Euclid's treatment of the limiting theorems in Book XII adheres to the 
methods actually used by Eudoxus. 

The principal advantage which Euclid could have obtained by employing this 
reconstructed alternative method for the theorem on circles is its elimination of 
the need to hypothesize the existence of the magnitudes X and Y satisfying the 
proportionalities A:X = C:D = Y:B. This hypothesis of the existence of 
the fourth proportional is a fixture in all applications of the Eudoxcan-Archi- 
mcdcan method of limits, as well as in the reconstructed Eudoxean proportion 
theory mentioned above. It is one of several striking instances of the appeal to 
nonconstructive assumptions within the ancient geometry/ 15 Like Bryson's in¬ 
termediate square, these magnitudes can be assumed to exist via appeal to some 
intuition of the nature of continuous magnitude in general. Here one perceives 
no concern whatever to ground such appeals on explicit constructions of the 
hypothesized terms. Indeed, Philoponus' remark on the circle quadrature entirely 
separates the issue of the existence of the square equal to the circle from that of 
the construction of (hat square, in (hat the former is assumed from (he start as 
the basis for the latter. Ever since Zeuthen, the view that constructions serve 
the purpose of existence proofs has been a favorite interpretation among writers 
on ancient geometry/ 9 The present counterinstanccs should give one pause in 
subscribing to this view as if it were one the ancients would have held of their 
own efforts. 

Among the ancient constructions of the circle quadrature, the method utilizing 
the special curve called the “quadratrix" is often assigned a pre-Euclidean origin. 
Pappus cites its use by Menaechmus* brother Dinostratus and by Nicomedes 
(late 3rd century B.c), and the latter is affirmed by lamblichus. 90 But in two 
places Proclus associates this curve with a geometer named Hippias. 91 If this is 
indeed Hippias of Elis, (he Sophist, a contemporary of Socrates, (he quadratrix 
would be the earliest known instance of a special curve among the ancients, and 
Hippias would deserve to be ranked foremost in mathematical expertise not only 
among the Sophists of his generation, but among the geometers as well. 92 But 
the technical sophistication of this would then so strongly foreshadow discoveries 
by Eudoxus and Archimedes that one is immediately put on guard. A too hasty 
acceptance of its attribution to such an early figure threatens to distort one's 


natural view of the chronology of the ancient technical methods by as much as 
a century or two. 

Left to a vote count among modem authorities, the attribution would win 
acceptance comfortably. Tannery and Bjombo defended the case quite insistently, 
and it has been adopted by Cantor, Heiberg, Heath, and many others since/' 
Dissident voices include Allman and Hankel among the older writers and Guthrie 
more recently. 94 The case rests entirely on the fact that Proclus docs mention 
Hippias of Elis in one other passage, as an authority to support the claim that 
“Mamcrcus, the brother of the poet Stesichoms... had a reputation in geometry" 
in the time before Pythagoras. 95 Clearly, the context has nothing to do with the 
quadratrix and signifies nothing of the mathematical expertise of Hippias himself. 
Docs this single passage then justify shifting the burden of proof onto those who 
doubt the identification of Hippias the sophist with Hippias the geometer? The 
substantial body of fragments from the Sophist give no evidence of interest or 
competence in technical matters. 96 To be sure, Plato has Socrates, in the dialogue 
Hippias Minor , address him as "a skillful calculator and arithmetician.. .the 
wisest and ablest of men in these matters," to which Hippias assents quite 
unequivocally; he is later noted as adept in geometry and astronomy as well, 
and indeed boasts of his own skill in all the arts, including the making of rings, 
seals, the very shoes, cloak, and tunic he was then wearing, not to omit the arts 
of memory, of writing prose and poetry of all forms, of music and so on/ 7 The 
point is surely clear: Hippias is being ribbed as a self-styled jack of all trades, 
but an actual master of none in particular. This is slippery ground to base a 
claim of his accomplishments in the field of geometry. Can one so readily 
disregard the negative testimony of Aristotle, who cites the circle quadratures 
by Hippocrates, Antiphon, and Bryson, but none by Hippias? The later Aris¬ 
totelians, for all their ample commentary on these passages on the circle quad¬ 
rature, likewise omit any mention of Hippias. Yet we may infer from Proclus 
that this man's contribution was hardly a casual one: 

Apollonius proved the properties of each of the conic lines, and Nicomedes (did 
the same) for the conchoids, and Hippias for the quadratriccs, and Perseus for the 
spirits [i.c., the sections of the torus).** 

Thus named in the company of late 3rd-century figures like Apollonius and 
Nicomedes, known for entire treatises devoted to the properties of the curves 
associated with them, Hippias would appear to be credited here for a comparable 
treatment of a class of curves, the “quadratrices" (note the plural). Yet several 
writers, including Proclus himself, assign a share in the study of this curve to 
Nicomedes also; indeed, it was Nicomedes, according to Iambiichus, who ap¬ 
pears to be responsible for naming the curve “quadratrix" (tetragonizousa) in 
recognition of its role in the quadrature of the circle. One wonders what was 
left for Nicomedes to do, however, if Hippias the Sophist had so advanced the 
study of this curve more than two centuries earlier. When we consider further 
that the name Hippias was quite common, 99 how can we deny that the Hippias 


82 


Ancient Tradition of Geometric Problems 


The Geometers in Plato’s Academy 83 



Figure 16 

of the quadratrix was not the 5th-century sophist, but rather a 3rd- or 2nd-ccntury 
geometer extending the work of Nicomedes? 

Thus, strictly on the basis of textual evidence, the case for assigning this 
curve to the sophist is exposed as implausible. But if the technical implications 
of this attribution had been considered by its advocates, it would never have 
been entertained seriously in the first place. We shall now find it useful to provide 
a brief account of this curve, in order to show how it relates to 3rd-century 
elaborations of the Eudoxean methods.' 00 With reference to the square ABGD, 
let us conceive that the ray AB rotates on A from an initial position at AB to a 
terminal position at AD, while in the same time BG moves parallel to itself from 
BG to AD (see Fig. 16). The moving lines will thus intersect at a point 0 which 
traces a curve running from B to a point K lying on AD. One may show that 
the given length AB is the mean proportional between the terminal length AK 
and the length of the arc of the quadrant BD. Since one can construct the line 
which is the third proportional of the lines AK, AB, 10 ' this line will equal the 
quadrant BD. Having thus rectified the arc of the quadrant, one can effect its 
quadrature, since its area is equal to one-half the product of its radius AB and 
its arc BD. ,C,J 

As one can well anticipate, the full proof of this construction, such as one 
finds it in Pappus, depends heavily on the use of limiting methods of the Eudoxean 
type. 1 ”' It is set out in the indirect form and assumes properties of similar arcs 
whose formal proofs may be associated with Archimedes. 104 A certain subtlety 
of method is required even to secure the conception of the terminal position at 
K; for at this position the defining rays coincide. This need for an appeal to 
continuity to specify K occasioned misgivings over the method among later 
commentators like Spores. 105 One can hardly imagine that it would have failed 
to spark discussion among 4th-century geometers and philosophers and that some 


reflection of this would emerge in the comments by Aristotle and his interpreters. 
One further notes that the method actually effects a rectification of the arc BD. 
Yet Aristotle is quite ignorant of any such method, when in the Physics (VII, 
4) he criticizes a more primitive method of the following sort: 

What then is the situation (with respect to motions) in the case of the circle and 
the straight line? For it would be absurd to suppose that this cannot be moved in 
a straight line similarly to that in a circle, but that the straight must necessarily 
move either faster or slower .... Nor docs it matter at all if one were to say that 
the straight docs in fact necessarily move faster or slower. For then the arc shall 
be greater and less than the straight line, so that also (it shall be) equal For if in 
the time A the one traversed the (line) B. the other the (arc) G. then B would be 
greater than G. For that is how “faster" was defined. Then also the faster (shall 
traverse) the equal in a lesser (time), so that there shall be sonic pan of (time) A 
in which that (moving along) B shall traverse the equal of the circle, while that 
(moving along) G (shall traverse) G in the whole time A. To be sure. then, if these 
arc compatible {symbltta)."* what has just been said will follow, namely, a straight 
line shall be equal to a circle. But they arc not compatible; hence, neither arc the 
motions. 

At the heart of this argument is the Brysonian principle: that if a varying mag¬ 
nitude can become now greater, now less than a specified value, at some time 
in the interval it will equal that value. Here, the inequalities arc arranged through 
the assumption that circular and rectilinear motions can be compared to each 
other via the relational terms “faster" and “slower. 1 It is hardly an advertise¬ 
ment of Aristotle’s geometric insight that he dismisses so perfunctorily this 
perfectly plausible rationale for the existence of the line equal to the circle. But 
one may note that Aristotle invariably treats with great respect the technical 
findings of such colleagues as Eudoxus and his followers. 10 * He would hardly 
have set aside so casually the present argument, were its conclusion and the 
kinematic conceptions supporting it already secured through the quadratrix con¬ 
struction by Dinostratus, let alone by Hippias over a half-ccntury earlier. 

The lemma associating the rectification of a circular arc with the area of its 
sector is indispensable if one is to derive from the quadratrix a solution of the 
circle quadrature. A proof is extant as the first proposition of Archimedes* 
Dimension of the Circle , and Pappus cites Archimedes explicitly both in this 
passage on the quadratrix and in numerous other places in the Collection where 
this same lemma is introduced. Other writers like Hero, Thcon, Proclus, and 
Eutocius invariably assign this result to Archimedes, often with specific reference 
to the Dimension of the Circle.' 09 We have no evidence whatsoever to justify 
assuming an awareness of this result by an earlier Greek geometer. This must 
certainly rule out any idea of its use by Hippias a century and a half before 
Archimedes. But it also discourages assigning its use by Dinostratus, around 
three-quarters of a century before Archimedes, for the manner of proof adopted 
by Archimedes involves only such techniques as were available to any geometer 
in the Eudoxean tradition." 0 For instance, he uses the familiar bisection method 
in the context of two independent applications of one-sided convergence, first 



84 Ancient Tradition of Geometric Problems 



A L PK D 


Figure 17 

by inscribed regular polygons, then by circumscribed polygons. None of the 
more sophisticated limiting methods, such as appear in his Sphere and Cylinder, 
arc brought to bear on the circle measurement. Can one comfortably accept the 
implication that pre-Euclidean geometers freely admitted this lemma in advanced 
constructions, like that involving the quadratrix, but that its elementary proof 
was overlooked for decades until Archimedes? 

These considerations thus reveal the utter implausibility of assigning the use 
of the quadratrix to a Presocratic like Hippias of Elis. But they raise the question 
of reconciling a post-Archimedean dating with the testimony of Pappus that “for 
the quadrature of the circle a certain curve has been received by Dinostratus and 
Nicomedes and some other more recent figures; it takes its name from the property 
relating to it, for it is called by them ‘quadratrix*/* 111 As we have already noted, 
lamblichus associated this naming of the curve with Nicomedes alone, but assigns 
no involvement on the part of Dinostratus. This suggests that Dinostratus might 
have introduced this curve in the context of a problem other than the circle 
quadrature. Pappus shows in detail how the quadratrix may be used for the 
division of an angle/ 12 Let us propose to divide a given angle into three equal 
parts. We introduce the quadratrix B0K and draw the ray AH making with AD 
an angle equal to the given (see Fig. 17). Let AH meet the curve at 0 and draw 
the ordinate 0L; next divide 0L at M such that ML is its third part. If a perpen¬ 
dicular to 0L is drawn through M, it will meet the curve at a point, say N. Then 
the ray AN makes with AD an angle one-third that of the given angle HAD. 
This is evident at once from the motions which define the curve; for they entail 
that the ordinates have the same ratio as thd associated arcs, e.g., that 0L : NP 
= arc HD : arc QD, hence, also, as the associated angles (HAD, QAD). 

It is not difficult to imagine how a curve of the above sort might be introduced 
for the purposes of the angle division. Aristot!e*s critique of (he attempted 


The Geometers in Plato's Academy 85 

rectification of the arc suggests an extremely simple practical procedure: figure 
the given angle as HAD in a circle, and with a cord or flexible ruler lay out the 
arc HD; next straighten the ruler and divide the segment HD in the desired ratio 
(say, QD : DH = I : 3); now lay the ruler back against the circle to find the 
arc QD, whence angle QAD will be the desired part of HAD. Thus, the angle 
division brings forward a natural association of curvilinear and rectilinear mag¬ 
nitudes. Further consideration might lead one to try to correlate circular and 
linear motions, the one traversing arc BD uniformly in the same time that the 
other traverses the straight line equal to the arc BD, and then to replace the latter 
with the precisely similar motion traversing the straight line BA. Although it 
would hardly be a trivial step to contrive in (his way a curve answering these 
motions and work out its application for the angle division, the way has been 
well prepared through the kinematic constructions of curves by Archytas and 
Eudoxus. In this way Dinostratus was in the position to convert the simple 
procedure, too hastily dismissed by Aristotle, into an ingenious method based 
on a new curve. A slightly different orientation of the same motions yields the 
spiral studied somewhat later by Archimedes and his colleague Conon/ 1 ' One 
would thus suppose that their discoveries relating to the spiral led Nicomedes 
to discover the use of the former curve for the circle quadrature, and so to dub 
it the “quadratrix/* Only after him, around the beginning of the 2nd century 
o.c at the earliest, could the geometer Hippias compile these findings into a 
writing on this curve. 

This view of the origins and study of the quadratrix charges Pappus with an 
erroneous, or at least misleading, statement in his giving to Dinostratus a share 
in the use of this curve for circle quadrature. But he is hardly free of similar 
inaccuracies elsewhere. For instance, he writes thus of the early efforts on angle 
trisection 114 : 

Having set out to divide the given rectilinear angle into three equal parts, the 
ancient geometers were at a loss for the following reason. We say that there arc 
three kinds of problems in geometry: some arc called planar, some solid, and some 
linear. (Pappus now explicates this classification in detail ! Since there is this 
difference among problems, the former geometers were hot able to discover (a 
solution for) the cited problem of the angle, being by nature of the solid class, 
when they sought it by means of planes."’ For the sections of the cone were not 
yet familiar to them, and so they were at a loss. Later, however, they trisected 
the angle by means of the conics, using for its discovery the following neusis. 

Pappus* thinking here is apparent. The angle trisection is assuredly a solid 
problem, that is, one solvable by means of conic sections. Hence, its proper 
solution would elude geometers until they had access to the requisite body of 
theorems on conics. The method he goes on to discuss betrays strong signs of 
a provenance near the lime of Apollonius, however, so that Pappus might be 
construed to mean that no earlier methods of angle trisection had been worked 
out. This, of course, is an inference which no one would accept. Methods using 
the “quadratrix/* the spirals or the neuses of Archimedes, or the conchoids of 
Nicomedes would all be assignable to periods earlier than the end of the 3rd 


86 Ancient Tradition of Geometric Problems 

century. Pappus has allowed his use of the conceptions and terminology of the 
later development of geometry to distort his characterization of the earlier efforts, 
much as we have argued Eratosthenes did in alluding to the “cone sectioning*’ j 

of Menaechmus. But if indeed Pappus supposed that the mere introduction of a 
certain curve, as by Dinostratus, carried with it a knowledge of the whole range 
of its applications, embracing both the solution of the angle trisection and that 
of the circle quadrature, wc need hardly imitate his mistake. For it is surely 
plain that the latter effort at least depended on advances in geometry due to ! 

Archimedes. j 

GEOMETRY AND PHILOSOPHY IN THE 4TH CENTURY 

The technical methods of geometry experienced significant advances at the hands 
of the group of geometers affiliated with Plato’s Academy. Problems like the 
cube duplication and the angle trisection were attacked and solved in a variety 
of ways. Special curves were devised for these ends via ingenious intuitions of 
mechanical motions. Archytas, for instance, used the curve formed as the in¬ 
tersection of a torus and a cylinder for doubling the cube; of the same kind was 
the “curved line” of Eudoxus, arguably the ophiuridc generated by the moving 
lines in the pseudo-Platonic solution; so also was Dinostratus’ quadratrix, con¬ 
ceived for trisecting the angle. At the same time the techniques of the application 
of areas were being implemented for the solution of a range of problems, in¬ 
cluding the construction of regular polygons and solids. Menaechmus perceived 
how these give rise to second-order curves by which the cube duplication can 
be solved, although his method for producing the parabolas and hyperbola so 
defined is more likely to have been a pointwisc procedure than the conception 
of sectioning the cone familiar to Euclid and his contemporaries several decades 
later. Furthermore, the method of analysis received considerable development 
in this period, exploited not only by geometers like Lcodamas, Leon, Eudoxus, 
and Menaechmus, but even by a philosopher like Aristotle in his discussions of 
problem solving in general. 

Responding to the philosophical environment of their efforts, these geometers 
markedly improved the formal basis of the organization and proof of geometric 
theorems. Proclus names about a dozen figures, most having an association with 
the Academy, who contributed to the composition of Elements of geometry, 
precursors of the various parts of Euclid’s treatise. 1,6 Particularly noteworthy 
were the insights t>y Eudoxus, who instituted techniques of limits and of pro¬ 
portions in a manner exemplary for its formal precision. The former filled gaps 
in Hippocrates’ treatment of the measurement of the circle, showing how his 
use of infinite limiting procedures could be effected through finite sequences in 
the context of indirect proofs. Eudoxus’ theory of proportions permitted the 
extension of theorems on proportions to the cases of incommensurable magni¬ 
tudes, and so secured the technical advances in the study of irrationals, most 
notably by Thcaetetus in the preceding generation. 117 One thus sees that Eudoxus, 
however imbued with the spirit of formalism encouraged by his philosopher 


The Geometers in Plato's Academy 87 

colleagues in the Academy, nevertheless drew the principal inspiration for his 
technical efforts from the earlier geometers like Hippocrates, Archytas, and 
Theaetetus. 

This convergence of geometric and philosophical interests has sorely tempted 
many scholars engaged in the study of these ancient efforts. Accordingly, no 
other period of the ancient geometry has been so thoroughly laced with inap¬ 
propriate interpretive positions as this pre-Euclidean period. 1111 Concerning the 
early circle quadratures, one commonly attributes to Sophists like Antiphon and 
Bryson pioneering insights into the nature of limiting methods, while to Hip¬ 
pocrates one imputes blatant oversights and elementary logical errors. 11 * De¬ 
mocritus emerges as a precursor of Archimedes in the devising of a geometric 
method of indivisibles, 120 while Hippias of Elis is alleged to anticipate his dis¬ 
coveries on the measurement of the circle through the use of the quadratrix. 
Eudoxus is motivated not by the ambition to advance the technical achievements 
of earlier geometers, but rather by the concern to accommodate the arguments 
of Zeno and Aristotle against the use of infinites. 121 Indeed, dialectical motives 
like the search for existence proofs arc read into the whole ancient activity of 
solving geometric problems of construction. 

Whence this tendency to distort the interpretation of the ancient geometry? 
Reconstructing the motives of others, even of near contemporaries, is always 
an uncertain enterprise. But I suspect, in this instance, that the particular intel¬ 
lectual background of the principal commentators is a significant factor. Tannery 
and Bjombo, for instance, those most responsible for the wide acceptance of 
the case relating to Hippias of Elis, are remembered primarily for their com¬ 
petence in philosophy and philology. How natural that they should seize upon 
the notion that a man of their own mind, like this Sophist, could have influenced 
the development of geometry! By contrast, no such presumption marked the 
view of Hankel, by profession a mathematician. One observes, furthermore, that 
the period of the late 19th and early 20th centuries witnessed a phenomenal shift 
in perspective on the nature of mathematics, 122 Developments in analysis, ge¬ 
ometry, number theory, and the theory of sets, for instance, revealed that certain 
forma] issues, long ignored or unperceived, lay at the heart of important technical 
difficulties, and one might even maintain that the whole of mathematics was 
subsumed within logic. Hard on the heets of the satisfactory resolution of foun¬ 
dational questions in analysis and geometry, however, came a new “foundations 
crisis” in the field of logic itself. In all this, the close interaction of mathematical 
and philosophical inquiries was evident as a stimulating area for research. From 
this vantage point one could readily project a comparable form of interaction 
between the philosophical paradoxes raised by Parmenides and Zeno and the 
geometry of the early Pythagoreans; or between the philosophical repercussions 
of the discovery of irrationals and the choices of technical methods by the 5th- 
and 4th-century geometers. One hypothesized a paralysis of research, a renun¬ 
ciation of the use of proportions in geometry, and so on, until this impasse was 
finally broken through the efforts of Eudoxus. 123 One seemed not to care that 
the extant evidence, fragmentary as it is, indicates no signs of such paralysis or 



88 Ancient Tradition of Geometric Problems 


The Geometers in Plato's Academy 89 


renunciation, but rather a remarkable degree of continuity in the development 
of technical methods. 

The alternative view advocated here acknowledges the significant interaction 
between geometry and philosophy over the special field dealing with the forma! 
nature of proof, but that, on the whole, the two disciplines developed autono¬ 
mously. Is that remarkably different from the course of modem mathematics? 
Save for those actively engaged in the study of foundational questions, how 
many mathematicians accept the general implications of those studies as an 
immediate factor in their own researches? It is interesting to consider the case 
of O. Bcckcr. a prominent interpreter of pre-Euclidean geometry. As an adherent 
to the Husserlian phenomenology in the 1930’s, Becker was especially sensitive 
to the questions of method raised by the earlier mathematical intuitionists. He 
thus probed the work of Eudoxus for signs of a similar concern. 124 If he did 
indeed discern formal aspects of the development of proportion theory never so 
clearly perceived by scholars before him, he nevertheless failed to find evidence 
of other concerns of the modem philosophy of mathematics, like the stricture 
against nonconstructive assumptions in proofs. But how many of Becker’s own 
contemporaries shared these concerns in their researches? 

This habit of reading present views into the past is hardly a recent phenom¬ 
enon. The later Pythagoreans told the tale of one Hippasus doomed to shipwreck 
for his sacrilege of having divulged the secret of the irrational to the uninitiated. 125 
The story of the Delian oracle seeks to link Plato directly with the early researches 
on cube duplication, and Eratosthenes converts it into a parable of Plato’s phi¬ 
losophy of mathematics. 126 Here is material for theories of “foundations crises” 
and philosophers* influence on the progress of geometry, should anyone seek it. 
Let the ancients have their fun, spinning yams to illustrate their favorite notions 
in philosophy. But we need not take these too seriously as guidelines for tracing 
the history of the ancient technical researches. 

NOTES TO CHAPTER 3 

1 In Euclidem . p. 66. 

: On these geometers, see my Evolution of the Euclidean Elements. Ch. 3, 7, 8. 
Short accounts may be found in the Dictionary of Scientific Biography. 

1 Perhaps the most ambitious attempts to assess the mathematical component of 
Aristotle’s thought arc T. Heath. Mathematics in Aristotle and H. J. Waschkicss, Von 
Eudoxos zu Ar is to teles. 

4 Cited by Eutocius in Archimedes, Opera . ed. Heiberg (2nd ed.). HI. p. 96. 

* Ibid pp. 84-88. An extremely lucid account is given by B. L. van dcr Wacrdcn. 
Science Awakening . pp. 150 f. See also O. Becker. Mathematisches Denken der Ant ike. 
pp. 76-80 and T. L. Heath. History of Greek Mathematics. I, pp. 246-249. 

6 Becker ( op . r/7., p. 79) presents this curve via a somewhat modified construction 
and cites its use by Villapaudo, as reported by Viviani (1647); see also G. Loria. Ebene 
Kurven , 1902. p. 317. Kepler's use of the same curve is noted by W. Breidcnbach, Das 
delische Problem. Stuttgart, 1953. pp. 31 f. 


7 For this part of the analysis, one may compare the account by van der Wacrden. 
The principal difference lies in that I seek a determination of the locus of point I as a 
result of the analysis, while van der Waerden introduces it as an unexplained intuitive 

, insight at the start. 

* Eutocius, op. nr., p. 56. 

! 9 See. for instance. Heath, History. I, p. 249; and l. Thomas, Greek Mathematical 

Works , I. p. 260n. 

10 P. Tannery, "Ligncs courbcs,” Bulletin des sciences mathtmatiques. 1884, 19, 
p. 101; cf. also Me moires de la Sociiti des sciences physiques et naturelles de Bordeaux. 
2j, pp. 277-283. Accounts appear in Heath, History , I, pp. 249-251; and Bcckcr, op. 
ci/., pp. 78-80. Breidcnbach also notes Descartes* use of this curve (op. cit.. pp. 29 0; 

• cf. the Gtomttrie (ed. Smith and Latham), pp. 154-157. 

11 One may note that the curve generated via the condition AR = AS [Fig. 6(a)) is 
extendable beyond the range of the intersection of the torus and the cone (i.e., up to the 
point where AR = AB). Setting PS = y, AS = x, and AN - a, one obtains y : x = 
Vjc 2 - cf a, or aV = x*(x 2 -a 1 ). The curve extends to infinity with increasing slope 
as x increases, becoming asymptotic to the parabola ay - x*. Its inflection point V lies 
at the intersection of the line x = Aa = VYj a and the circle of diameter Ac = Y 2 VY 2 
a , so that AV = Y> a. (Fig. 6(b)) V here corresponds to I in the construction of the two 
mean proportionals AK, AI between AD, AB, for AD : AB = V3V3 : V2V2. For 
Ab : AN = 3 : 2, V will correspond to T. the terminus of intersection of cone and torus. 
The problem of doubling the cube corresponds to Ac : AN = 4:1. 

12 R.C. Riddell, "Eudoxan Mathematics and the Eudoxan Spheres,” Archive for 
History of Exact Sciences, 1979, 20, pp. 1-19. 

n In the modern theory of curves, this is termed the "spherical Icmniscatc"; projected 
onto the plane tangent to the cylinder at (he double point, it becomes the Bemou Ilian 
Icmniscatc; cf. C. Zwikker. Advanced Plane Geometry'. Ch. XVII. For its relation to the 
sections of the torus, see Chapter 6. 

14 This sketch is not intended to be self-sufficient. For details, consult Riddell. Other 
treatments arc given by O. Ncugcbauer, Exact Sciences in Antiquity, pp. 153 f, 182 f; 
van dcr Wacrden, op. cit., pp. 180-182; and Heath, History , I, pp. 329-335, 

** Detailed critiques of Eudoxus' astronomical theory are given by Ncugebaucr. 
History of Ancient Mathematical Astronomy, pp. 677-685: Heath, Aristarchus. Ch. 16; 
and L. E. J. Dreyer, History of Astronomy .... Ch. IV. 

“ Cf. Proctus, in Euclidem, pp. 112, 127 f. 

17 Drawings of this curve arc provided by Riddell, op. cit.. Figs. 12, 14. 

“ Under this condition, the projection curve is a two-cusped epicycloid, as shown 
in Fig. 8(b), and is related to the epicyclic curves discussed in Chapter 7. When 0G is 
less than GQ, the curve will have two loops in place of the cusps, and when 0G is greater 
than GQ, the curve will arch smoothly inward at these positions. Riddell's Fig. 11 
illustrates a curve of the latter form. 

19 This is Riddell's solution method. But it leads to an alternative which dispenses 
with reference to the rotating spheres. If we conceive of the plane curve as generated by 
the epicyclic motion of circle JRH, then the perpendicular from T will meet it in R, while 

! the generating circle will have assumed the position JRH; thus, 0J is determined, from 

• 

i 




90 Ancient Tradition of Geometric Problems 


i 


which the construction of OS. OV follows. If we admit the epicyclic curve alone, without 
reference to the continually changing positions of circle IRH. nevertheless, the specific 
position of that circle corresponding to the point of intersection R can be produced by a 
planar construction; for the circle will pass through a given point and be tangent to two 
given circles. The latter problem falls under one of the classes of planar constructions 
treated by Apollonius in his Tangencies: cf. Chapter 7. 

" But one n0,e ,hat ,he modified Eudoxean scheme which Schiaparelli would 
assign to Callippus introduces (he sphere of double speed, as in Riddell's construction 
The three-sphere arrangement, however, produces a more complicated curve in which 
the figurc-8 obtains additional double loops at each extremity. For a discussion see 
Heath. Aristarchus, pp. 212-216. 

" Eutocius, op. cit., pp. 56-58. 

” Sec ,he <l'M<*ssion in Chapter 2. According to the lexicon of Suidas. Eratosthenes 
was known as "a second ( deuteros ) or new (neos) Plato”; cf. I. Thomas, op. cit.. II. 
p. 260. 

” See. in particular, Republic 510 c-c, 525 ff. 

M This is comparable to the view proposed by van der Waerden. op cit., pp. 159- 
165. He does not actually attribute the pseudo-Platonic construction to Eratosthenes; but 
that appears to be implied in his account. 

* Cf. Lona, Ebene Kurven, I, p. 50. A more detailed discussion appears in Chanter 
6 (note 107). K 

H Banu Musi, Verba filiorum. Prop. 17; cf. M. Clagctt. Archimedes in the Middle 
Ages. I, pp. 340-344. Further discussion will appear in the sequel. Only the geometric 
form of the construction is given in the Latin version of this work (made by Gerard of 
Cremona in the I2lh century). But the Arabic version, extant in the recension by al-JOsi 
(13th century), adds to this the details of the physical mechanism by which the construction 
can be produced. 

Verba filiorum, Prop. 16; cf. Clagctt, op. cit., pp. 334-340. 

” N ° ,c - for ins,ance - Ih e expressions "to the extent that... ” (epi tosouton mechris 
an) and the beam has the position such as has.. ( ton kanona thesin echein hoian 

cchei. ) used here by Eutocius (op. cit.. p. 58) and their reappearance in his versions 
of the methods of Philo and Apollonius (ibid. . pp. 62.64,66). His mechanical terminology 
parallels that in his texts of Nicomedes (ibid., pp. 98-100) and of Pappus (ibid., p. 70; 
cf. Collection VIII. p. 1070). less (hat in his text of Eratosthenes (ibid., pp. 92-94). 

** Eutocius. op. cit.. pp. 78-84. 

See. for instance. H. G. Zeuthen. Lehre von den Kegelschnitten. Ch. 21, especially 
pp. 457-466 and the synopsis of "Menaechmus' probable procedure" by Heath Historv 
I. pp. 251-255 and II. pp. 110-116 and his Apollonius, Ch. I. 

“ Af,cr sketching Zeuthen’s reconstruction of a Menacchmean derivation of the 
asymptote properly of the hyperbola. Coolidge observes: "I confess to finding it a bit 
difficult to believe that Menaechmus could work out all this. But we are forced to accept 
something or the sort if we are to credit him with the discovery of the conic sections in 
connexion with the problem of duplicating (he cube" (History of the Conic Sections, p. 
5). Such rare, but well-founded misgivings are set at rest in the view proposed below. 

” Both relate to the elliptical appearance of circles viewed obliquely. For 
sec Chapter 4. 


The Geometers in Plato's Academy 91 

M For accounts of the early theory, see Zeuthen. op. cit.. Ch. 2. 19. 20. and Heath. 
Apollonius . Ch. 3. A compilation of Archimedean theorems on conics is given by Heiberg 
in "Die Kenntnissc dcs Archimedes iiber die Kegelschnittc.** Zeitschrift fur Mathematik 
und Physik, hist.-lilt. Abth.. 1880. pp. 41-67. 

u Eutocius. op. cit.. p. 96. 

" Note that ihc curves discussed above as reconstructions proposed for Eudoxus* 
method rely on pointwisc procedures, as do several of the accounts of other curves, like 
the quadratrix; cf. van dcr Waerden, op. cit.. p. 192. 

* Sec Diodes . cd. Toorncr. Props. 12 and 4, respectively. Eutocius {op. cit.. pp. 
66-70) presents a version of the former in which the points arc joined by rectilinear 
elements, rather than by means of a flexible ruler. But the flexible ruler is clearly indicated 
in Diodes' own account; cf. Toomer. pp. 159 f. 

M Eutocius in Apollonius, Opera, cd. Heiberg. II. pp. 230-234. 

See Heiberg. Mathematici Graeci Minores. 1927. pp. 78 ff. 85 ff. These con¬ 
structions arc discussed in Chapter 6. 

w Diodes, op. cit.. Prop. 10. 

m Proclus, In Euclidem . p. 419. Plutarch. M or alia. 720 a. 1094 b; sec Heath. History. 
L pp. 144 f. For accounts of this technique, sec Heath, ibid. . pp. 150-154. van dcr 
Waerden, op. cit,. pp. 118-126; and Becker, op. cit.. pp. 60-64. 

41 In Fig. 13(a). if CD = X, DF = A. E bisects DF. and DH = B, then the 
"hyperbolic** case makes DH the mean proportional between CD and DG ( = A + X). 
This gives rise to the right triangle EDH having two given sides ED ( = •/; A) and DH; 
thus, EH ( = Vi A + X) is also given, from which CD can be solved. Similarly, in Fig 
13(b) the "elliptic** case makes DH the mean proportional between CD and DF. whence 
ED ( = */j A - X) is given from the triangle EDH. For the relation between these analyses 
and the syntheses given by Euclid, sec Heath. Euclid’s Elements. I. pp. 382-388 and II. 
pp. 260-267; and Becker, op. cit., pp. 62 f. 

42 See Neugcbaucr, Mathematical Cuneiform Texts . 1945. and his Exact Sciences in 
Antiquity, pp, 40-42; examples are given by van dcr Waerden, op. cit., pp. 63-71. and 
their relation to the Greek technique is elaborated, ibid., pp. 122-124. 

° On the manner of this transmission, sec my "Techniques of Fractions." Ncuge- 
baucr remains vague [Exact Sciences , pp. 149-151. 181 0. although he insists on a real 
link between the two traditions. 

44 Collection (111). I, pp. 132-162. This is not to suggest that Pappus’ treatment has 
a pre-Euclidcan provenance. These constructions may well have their origin with Aristacus 
in the time of Apollonius; cf. Chapter 7. 

45 In Euclidem . p. 67. For an interpretation of this reference to Eudoxus, sec my 
Evolution, Ch. VIII. 

44 Sec Chapter 2. especially note 88. 

47 For further discussion, see my Evolution, Ch. II/ll, VI/IV. 

44 This results easily from an analysis based on “completion of the square.** Assuming 
BF 2 = FE • BE, one has BF 2 + BF • BE = BE 7 , or (BF + Vi BE)*’ *= BE : + ('/; 
BE) 2 , so that BF + Vj BE is the hypotenuse of a right triangle of sides BE, BE. 
Setting BG « Vi BE and HG = EG, it follows that BF = HB is the required length. 
The corresponding synthesis is given in Euclid's II. II. This form of analysis has been 


92 Ancient Tradition of Geometric Problems 

proposed as underlying some of the computations preserved in Mesopotamian tablets 
from the mid*2nd millennium b c ; cf. van dcr Wacrdcn. Science Awakening , pp. 69 f. 
It thus contributes to the view of the transmission of the older technique to the Greeks 
during the pre-Euclidean period cited in note 43 above. 

** R. Fischler proposes some further views on these early studies in “A Remark on 
Euclid II, II,** Historic Mathematical 1979, 6, pp. 418-422. 

One must avoid assigning too much weight to this form of negative testimony, 
however. Euclid's work is not an exhaustive repository of the geometric knowledge of 
his time, and one must allow for the prior availability of alternative methods from among 
which Euclid could choose for the proofs in the Elements (cf. my Evolution, Ch. I/ll). 
Nevertheless, it is always preferable, when engaging in reconstruction, to have some 
positive evidence from the surviving documents. 

Sec the discussion of the Aristotelian and Apollonian locus problems in Chapter 
4; the related problem preserved by Pappus (Chapter 4, note 16); and the neuses from 
Heraclitus and Apollonius (Chapter 7). 

'* Euclid adopts an oddly roundabout method here, taking angle EFA as an exterior 
angle to triangle CFA. One could proceed more directly by considering the triangles 
AFE, CAE which have equal angles at C and A and share the angle at E; thus, the 
remaining angles EFA, EAC arc equal. 

" Heath reports just such a reconstruction, taken from Todhuntcr; cf. Euclid’s E/e* 
ments , II. pp. 99 f. 

' 4 See also the translations by Heath. History, 1, p. 299; and Thomas, op . c/7., 1, 
pp. 394-397. 

" This was proposed by August (1829) and Butcher (1888) and adopted by Heath, 
History . I, pp. 299-303. For a detailed survey of these and many other proposals, sec 
R. S. Bluck, Plato*s Meno . 1961, pp. 441-461. Bluck discusses the general method in 
this passage in relation to geometrical analysis, ibid., pp. 76-85. 

* So Heijboer, cited by Bluck, op. r/7.. p. 448. Cf. Heath's response to the objections 
by Benccke. History , 1, p. 302. 

Bluck, op, r/7.. Introduction. Ch. E. csp. pp. 118*120 reviews the arguments for 
the dating of the Meno. 

w We provide an elementary proof that the equilateral triangle is the greatest of all 
triangles inscribed in the same circle. Since of all triangles on a given chord AB (Fig. 
19(a)) the isosceles triangle CAB is the greatest, we have only to compare the equilateral 
with the isosceles cases. First consider the isosceles triangle ADE |Fig. 19(b)) whose 
base DE is parallel to and shorter than a side BC of the inscribed equilateral triangle 
ABC. Since the bisector H of arc ADE lies on arc AB, triangle ABE will be greater than 
ADE; for the altitude BF of the former will be greater than the altitude DG of the latter. 
From the previous case, the equilateral triangle ABC will be greater than ABE. since the 
former is isosceles with respect to the common base AB. Hence, a fortiori, ABC is 
greater than ADE. Similarly, if side DE is greater than BC [Fig. 19(c)), ABE will again 
be greater than ADE, for here the bisector of arc ADE will lie on arc BC. It follows as 
before that the equilateral triangle ABC is greater than ADE. We have thus established 
that the equilateral triangle is greater than any other triangle inscribed in the same circle. 
It seems to me that no feature of this proof of the diorism would have posed any difficulty 
for a Greek geometer. Thus, the technical effort from which Plato draws here must surely 


The Geometers in Plato's Academy 93 



Figure 19 


94 Ancient Tradition of Geometric Problems 



I- X -1 

|- A -1 


(a) 


Figure 20 



(b) 


have been directed toward the explicit construction of the inscribed triangle of given area. 
Our good fortune in learning of this effort through this passage thus owes to Plato's slip 
in phrasing the diorism in accordance with the terms of the construction problem itself, 
thus losing sight of its independent treatment via elementary methods. 

** In Fig. 20(a), setting HG = Y. CE = A. and CG = X, we have, from the 
property of the tangent to the hyperbola. FH = HK, whence FG = GC (cf. Chapters 
5. 6, 7). Since HG is the mean proportional of FG, GD and also of EG. GC (= FG). 
it follows that DC = GE. so that X * Y* A. The property of the tangent can readily be 
deduced from the con slant-product property of the curve. Let the curve be figured with 
respect to the lines FC, CK such that for any points H, H' on it, GH • HD = G'H' * 
H'D' (Fig. 20(b)), and let the chord HH' meet CF. CK at F. K. respectively: it is claimed 
that FG = D'H\ For HD : H'D' = G'H' : GH; by similar triangles, G'H' : GH = FG' 
: FG. Thus. HD - H'D' : H'D' = FG' - FG : FG. Since HD - H'D' = FG' - FG 
{ = GG'). H'D' = FG, as claimed. If the secant is now adjusted so that H' approaches 
H. in the limiting position of the tangent one has FG = GC, as required above. One 
may note that this derivation effectively reverses the order followed in Apollonius' Conics , 
for there the constant-product property of the hyperbola (II. 12) follows after the equality 
of the segments of the secant, FH - H'K (II. 8). As it happens, Apollonius does not 
use II. 8 for the proof of the lemma II. 10 (that FH • HK « FH' * H'K) on which II. 
12 depends. This odd isolation of II, 8 (and its corollary tangent theorem. II, 9) doubtless 
signifies stages in the composition and transmission of the Conics; see Chapter 7. 

m in Eudidem. pp. 220 f. This passage is discussed by Heath, Euclid , 1. p. 128. 

M A similar instance is cited by Pappus as an example of an "indeterminate" ( adior - 
is ton) problem: the application with square excess over a given line segment yields widths 
arbitrarily large or small, depending on the magnitude of the area applied {Collection 
(VI). ||. p. 542). Thus, the "unordcrcd" problems of Amphinomus arc parallel to the 
"indeterminate" of Pappus, in that each refers to problems having separate solutions for 
each choice of a parameter from a specified range. Mcnaechmus’ curves for the cube 
duplication correspond precisely to this class of problems. 

** In Eudidem . p. 202. A puzzle here is that Aristotle does maintain that demon- 





The Geometers in Plato's Academy 95 

strations explain the causes of propositions; cf. Posterior Analytics 1. 2, 71 b 8 - 32. and 
24, 85 b 23 - 28. The difficulty is noted by G. Morrow. Proclus: A Commentary .... 
1970, p. 158 n. 

In Eudidem. pp. 77 f. There would seem to be a discrepancy with Proclus* 
ascription to Amphinomus of a classification of problems (see note 61 above). But perhaps 
a commitment to the priority of theorems over problems need not entail the view that 
the format of problems should be eliminated altogether. 

** 78 a 7 - 14; cf. also Prior Analytics It. 2-4. Other discussions of analytic method 
in the more general dialectical context appear in A lie. Eth. 1112 b 11 - 24 and Post . 4mi. 
I. 22. 84 a 6 ff. Such passages arc examined by J. Hintikka and U Remes. The Method 
of Analysis . Ch. 2, in the context of Pappus’ account of this method ( Collection. VII. 
Preface). 

M This passage correctly understands analysis as the examination of the deductive 
consequents of the hypothesis; other passages in Aristotle and later writers (see the 
references in note 64) attempt to construe it alternatively as a search for appropriate 
antecedents, but this confusing feature conflicts with the unanimous testimony of the 
mathematical evidence. It would also render superfluous any concern over convertibility, 
since the analysis (of antecedents) would of itself have produced the deductive sequence 
of the synthesis. I shall not attempt to press further these early passages on the nature of 
analysis, but the reader who wishes to do so may consult J. Barnes. 'Aristotle. Men- 
aechmus. and Circular Proof," Classical Quarterly . 1976. 26. pp. 278-292 for a more 
ambitious interpretation. 

“ Soph. Ref. xi and Prior Ana . I. 9. For discussion see Heath. Mathematics in 
Aristotle, pp. 47-50. 

* 7 Philoponus, In Ana. Post., ed. Wallies, pp. 111-114 cites views of Alexander. 
Ammonius, and Proclus concerning Bryson’s argument. Simplicius’ long passage on the 
circle quadrature docs not consider Bryson {In Physica . ed. Diels. I. pp. 53-69). 

** The specification of squares appears only in the comment by pseudo-Alexander 
{In Soph. EL. ed. Wallies, p. 90); on this basis one has been led to speculate whether 
Bryson intended that the square formed as the arithmetic or the geometric mean of the 
constructed squares ought to equal the circle. But in the paraphrase by Philoponus. 
Alexander considers an unrestricted "rectilinear figure." thus indicating a more sophis¬ 
ticated view. 

** So Proclus. as cited by Philoponus, op. cit.. pp. Ill f. Proclus objects to Alex¬ 
ander's interpretation on the grounds that it would make the fallacy of Bry son's argument 
the same as that of Antiphon’s: that the circle will coincide with the straight line. Proclus 
thus seems to have in mind the case where the inscribed and circumscribed figures arc 
so close to each other that the intermediate polygon becomes indistinguishable from either 
and hence coincides with the circle. This need not be rejected as a possible view of 
Bryson’s argument, even if Proclus himself goes on to propose an alternative. 

70 Op. cit.. pp. 112 f. The principle already appears in the Aristotelian Mechanics 
(847 b 25): that whatever changes from one extreme to the other (c.g.. from convex to 
concave, or from great to small) must sometime be in the mean state (e g., straight or 
equal). It may well be rooted in Academic discussions about the greater and lesser, the 
limit and the unlimited, such as in Plato's Philebus (23 c - 27 c). 

71 Ibid., p. 112. Cf. Proclus. op. cit.. p. 234. who observes that in the ease of the 


1 





96 Ancient Tradition of Geometric Problems 

homed angle, one docs not pass from the lesser to the greater through the equal; that is. 
no rectilinear angle equals the curvilinear. 

Mixed angles are introduced in the pseudo-Euclidean Catoptrics , Hero’s Catop¬ 
trics , Euclid’s Optics in the recension by Theon. and the Bobbio mathematical fragment 
(discussed in Chapter 6, note 99). For references, see Toomcr, Diodes , pp. 156 f and 
my "Geometry of Burning-Minors in Antiquity." 

" Op. cit .« p. 112. 

M Archimedes proves this in a more sophisticated manner in Sphere and Cylinder I, 
5. 6. 

" So Heath. Mathematics in Aristotle, pp. 48 f. Sec also Becker. "Eudoxos Studien 
II" and "HI" (Quellen and Studien . 1933 and 1936. respectively) and lan Mueller, 
"Aristotle and the Quadrature of the Circle" in Infinity and Continuity in Ancient and 
Medieval Thought . cd. N. Kretzmann. Mueller quite plausibty injects a kinematic clement 
into the conception of the sequence of squares. 

7h For instance, the existence of the fourth proportional is regularly assumed in the 
limiting theorems of Euclid and Archimedes, doubtless in agreement with Eudoxus* 
procedure (see discussion below). See also Becker. "Eudoxos-Studien II" and my "Ar¬ 
chimedes and the Prc-Euclidcan Proportion Theory." Mueller points to this as a char¬ 
acteristic nonconstructive feature in Euclid’s geometric methods; see his Philosophy of 
Mathematics ... in Euclid's Elements, 1981. Ch. 3.2, 6.3. 

” On Archimedes* use of Eudoxean techniques, see my "Archimedes and the Ele¬ 
ments** and "Archimedes and the Pre-Euclidean Proportion Theory." For accounts of 
Eudoxus* method of limits, sec van der Wacrden. Science Awakening, pp. 184-187 and 
Mueller. Philosophy of Mathematics, Ch. 6.3. 

T * Euclid proves this by considering the rectangle enclosing each segment on the 
sides of the inscribed polygon. Since the inscribed triangle equals exactly one-half the 
rectangle, while it. in its turn, is greater than the segment, the triangle is greater than 
one-half the segment. Hence, the passage from one polygon to the next will remove more 
than one-half the difference in area between the circle and the initial polygon. 

" This follows from the lemma (X, l) on the indefinite diminution of a finite mag¬ 
nitude through successive bisection. 

** Via this ploy, Euclid eliminates the need for circumscribed figures in all the limiting 
theorems in Book XII (i.e.. Props. 2. 5. 10. II. 12. 18). This feature may originate with 
Eudoxus* treatment of these theorems. But it is just the sort of refinement which one 
would expect from a subsequent rigorizing editor, for instance, a follower of Eudoxus 
or Euclid himself. One need not deny then that Eudoxus might have employed circum¬ 
scribed figures in his treatment. In this case. Archimedes could have seen in Eudoxus’ 
work a precedent for his own use of such figures, as in the Dimension of the Circle ; sec 
Chapter 5. 

•' Sec Chapter 2. 

B1 This was noted by Becker in his "Eudoxos-Studien IV" ( Quellen und Studien . 
1936). For a discussion of its implications for understanding Eudoxus’ form of proportion 
theory, see my "Archimedes and the Pre-Euclidean Proportion Theory." 

B1 Sec my "Archimedes and the Elements** and Chapter 5. 

M A more elaborate reconstruction is proposed by Becker in his "Eudoxos-Studien 


The Geometers in Plato’s Academy 97 


IV." For accounts of the Euclidean technique of proportions, see Heath. Euclid . 11, pp. 
116-131; van dcr Wacrdcn. op. cit pp. 187-189; Mueller, op. cit.. Ch. 3; and my 
Evolution of the Euclidean Elements , Ch. 8 and Appendix B. 

” One can readily provide a constructive test for the inequality A : B > n : m. If a 
sequence of regular polygons P is inscribed in circle A, while a sequence of similar 
regular polygons Q' is circumscribed about B, then one will eventually have P : Q' > 
n ; m , whence d fortiori A : B > n : m. A similar test applies in the case of the opposite 
inequality. 

16 This may be done by the bisection method, as in XII, 2, or by the alternative 
method of concentric circles used by Euclid for the volume of the sphere; cf. XU. 16. 

M This view is elaborated in my '‘Archimedes and the Pre-Euclidean Proportion 
Theory." 

M Sec Becker, "Eudoxos-Studien II." For other instances one may note Archimedes* 
neuses in Spiral Lines , and indeed all applications of neuses: also, the assumption of 
points of intersection of given lines and curves (cf. Heath, Euclid . I. pp. 234-240). 

" "Die gcomctrischc Construction als ‘Existcnzbcwcis’ in dcr antiken Geometric.** 
Mathematische Annaten, 1896, 47. pp. 222-228. For a criticism of this view in relation 
to the Euclidean geometry, sec Mueller, op. cit., pp, 15, 28. 

90 Collection (IV), I, pp. 250-252; tambtichus is cited by Simplicius, In Phvstca , I, 

p. 60. 

9t In Euclidem , pp. 272, 356. 

92 This is a view actually maintained by some; cf. K. Freeman, Pre-Socratic Phi¬ 
losophy, Oxford, 1946. p. 355; and W. D. Ross, "Hippias of Elis" in the Oxford Classical 
Dictionary . 2nd ed., p. 517. 

" Tannery. "Lignes courbcs," 1883. pp. 278-281; Bjdmbo, "Hippias," in Pauly 
Wissowa, 8, 1913, col. 1708 f; Heiberg, Geschichte der Mathematik ... im Altertum . 
p. 5; Heath, History , I, pp. 23, 225 f. Apparently, these writers maintained this as the 
*’traditional** view, and so felt little need actually to argue a textual case for it. Camor*s 
line is typical: this Hippias must be Hippias of Elis, for the ancient tradition preserves 
no mention of any other geometer of this name. But, of course, the tradition preserves 
merety single references to many geometers (as a glance at the name-indices to Pappus, 
Proctus, and Eutocius reveals); while wc may easily estimate that dozens have been 
forgotten altogether. Cantor also begs (he extremely important question as to whether the 
Sophist was a geometer, and so must invent reasons for his being omitted from Proclus* 
list of Euclid's precursors (In Euclidem , pp. 65-67). 

^ Hankel, Geschichte der Mathematik. 1874, p. 15 In. Allman initially accepts Han¬ 
kers view ( Greek Geometry , 1889. pp. 92-94), but is later persuaded by Cantor*s line 
and so adopts a position midway between Hankel and Tannery: that Hippias did indeed 
introduce the curve, but for angle trisection rather than circle quadrature (ibid., pp. 93n, 
189-193). This is the view held by van der Wacrdcn (op. cit., pp. 146, 191) and by 
Becker (op. cit. . pp. 95 ff): but Guthrie seems unconvinced ( History of Greek Philosophy. 
III. pp. 283 0. 

** In Euclidem , p. 65. 

* For a survey, see Guthrie, op. cit.. pp, 28(3-285. 

91 Hippias Minor 367 c-368 e. In the Protagoras (e.g.. 315 c, 318 e ) Hippias appears 






I lie ijctmiC'i. 


.* ■ »iiuum iiauiiiuii ui ucoincliic I'roblcms 

as a teacher of the mathematical sciences and natural philosophy* but no specific indication 
is given of his geometric expertise. 

** In Euclidem , p. 356. 

w Eighteen figures bearing this name are noted in Pauly Wissowa , Vol. 8. 

A more detailed discussion is given in Chapter 6. For other technical accounts* 
sec Heath, History . I, pp. 226-230; van dcr Waerdcn (on Dinostratus), op. cit., pp. 191- 
193; and Becker, Denken, pp. 95-98. 

,m Cf. Elements VI. II. 

'‘ ,2 Cf. Archimedes. Dimension of the Circle . Prop. 1. 

Collection (IV), 1, pp. 250-262; the proof is given in pp. 256-258. 

E.g,, that similar arcs have the ratio of the diameters of their respective circles. 
For a discussion of this theorem and its relation to the extant Dimension of the Circle , 
see my “Archimedes and the Elements" and “Archimedes and the Pre-Euclidcan Pro¬ 
portion Theory. Note that this use of the quadratrix also requires the equivalent of the 
inequality sin x < x < tan .r: the first known instance of this is in Archimedes* Dimension 
of the Circle , Prop. I. 

Cited by Pappus, Collection (IV), I, pp. 252-256. The difficulty is examined in 
the accounts by Heath and van dcr Waerdcn (sec note 100 above). 

"** From symbattein . asymbleta is often rendered as “incommensurable" (for which, 
however, the usual term is asymmetros ) or as “incomparable,” that is, literally, “unable 
to meet. Aristotle doubtless has in mind the faci that circles and lines cannot be made 
to coincide with each other. This would bring this passage into the context of Antiphon’s 
and Bryson's circle quadratures; cf. note 69 above and Chapter 2. 

Physics 248 a 19 - b 7; cf. Heath, Mathematics in Aristotle, pp. 140-142. The 
elliptical phrasing is quite typical of Aristotelian writing, but can be supplied with little 
fear of ambiguity. 

See, for instance, Aristotle's discussions of the Eudoxcan and Callippean planetary 
systems in Metaphysics XII, 8. 

"" Sec. for instance. Pappus, Collection (IV), I. p. 258. (V) pp. 312 ff, and (VIII), 
p. 1106; also In Ptolemaeum, cd. Rome. 1931, pp. 254 ff. Hero, Metrica 1. 26 (cf. I, 
37). Thcon, In Ptolemaeum , cd. Rome, 1936, pp. 359 ff (here following Zcnodoms; see 
Chapter 6). Proclus, In Euclidem , p. 423. One may of course add Eutocius* commentary 
on Dimension of the Circle, op. cit., pp. 228 ff. The writers and scholiasts in the metrical 
tradition growing out of Hero's work concur in the Archimedean attribution; cf. Hero, 
Opera, ed. Heiberg. IV-V. 

"" The elementary character of Archimedes' method here is one of the indications I 
employ for arguing an early dating for this work; see my “Archimedes and the Elements.** 

"* Collection (IV), I, pp. 250-252. Note that the word for “received** (from para - 
lambanein) would regularly mean “received [from )**, as in the manner of inheritance. I 
suggest reading “ apo** ("from”) for the manuscript's "hypo" (“by”), so to obtain “a 
curve received from Dinostratus et al.** This would have the effect of removing the 
implication made in the passage as it now stands, that Dinostratus introduced the curve 
in the context of circle quadrature. 

Collection (IV), I. pp. 284—286; the spiral is used for the same purpose, ibid., 
pp. 286-288. 





| 

J 


Uf See Archimedes* Spiral Lines and Pappus. Collection (IV), I. pp. 234 ff. This is 
discussed in Chapter 5. 

1,4 Collection (IV). I, pp. 270-272. 

,,J That is. by means of the “Euclidean" constructions using compass and straightedge 
alone. 

Mft In Euclidem , pp. 66 f. 

For a discussion of these studies, see my Evolution , Ch. 8. 

"• 1 examine several views of this sort in my “On the Early History of Axiomatics” 
and “Infinity and Continuity.** 

"• One perceives such a tendency in Heath's accounts and those he cites; see his 
History , 1, pp. 222, 224, I96n. 

120 The chief proponent of this view is S. Luria. “Die Infinitesimalthcoric dcr antiken 
Atomistcn*” Quelten undSrudien. 1933. B : 2. pp. 106-185; it is elaborated by J. Mau, 
Zum Problem des Infinitesimalen bei den antiken Atomisten. Berlin. 1954. Brief accounts 
appear in van der Waerdcn. op. cit.. pp. 137 f; and Guthrie, op. cit.. IL pp. 487 f. Sec 
also Heath. Euclid's Elements . III. pp. 366-368. 

1,1 The thorough study by H. J. Waschkicss of the roots of Aristotle's continuum 
theory {Von Eudoxos zu Aristoteles). I believe, pays insufficient attention to the technical 
sources of Eudoxus' geometric methods. By contrast. Mueller maintains that no specific 
philosophic motive need be invoked for understanding the methods of Eudoxus and Euclid; 
cf. op. cit., pp. 10, 234. 

122 For surveys of the background of the modem studies of foundations, one may 
consult N, Bourbaki, tlemenis d'hisroire des matMmatiques. "Fondcmcnts dcs mathd* 
matiques,” pp. 9-63; and M. Kline, Mathematical Thought from Ancient to Modern 
Times . 1972, Ch. 40-43.51. 

m For references, see my Evolution, Ch, 9 and the papers cited in note 118 above. 

124 Sec his “Eudoxos-Studicn I-1V,” Quellen und Studien, 1933-36. 

See my discussion in Evolution. Ch. 2. 

126 See Chapter 2. 






CHAPTER 4 


The 

Generation 

of 

Euclid 


With specific reference to flic three "classical” problems our sources preserve 
no solving efforts from the half-century or so separating the successors of Eu¬ 
doxus from Archimedes. Nevertheless, this intermediate period, the generation 
of Euclid in Alexandria, 1 witnessed important advances in the field of geometric 
problem solving. At least live works by Euclid alone bear directly on the im¬ 
plementation ot analytic efforts, while many of the more advanced problems 
which would concern geometers later in the 3rd century were defined and their 
analysis initiated at this time. 

Hie iJements of Euclid are justly described as the most widely known and 
used technical textbook ever written.* In this work Euclid gathered the separate 
fields of elementary plane and solid geometry and number theory and edited 
prior treatments to produce a comprehensive and generally uniform compilation. 
Here the efforts of Eudoxus and his disciples in the field of "exhaustion” 
measures of curvilinear figures are gathered, as well as their development of the 
theory of proportions of magnitudes; the researches of Thcactctus and his suc¬ 
cessors in the fields of irrational lines and the construction of the regular solids 
arc presented, together with the treatment of number theory on which they 
depend. These materials, which occupy somewhat more than half of the Ele¬ 
ments , constitute fairly well circumscribed subjects whose study is of interest in 
its own right. Hut the remainder of the work is more elementary in character, 
intended to provide the basis of theorems necessary for the further study of 
geometry. Throughout, the treatment is synthetic, so that it sometimes is not 
immediately evident what the significance of a particular theorem might be. This 


i ls ln,c * for instance, of much of Book II, where one meets theorems having the 
i tyfxttoncc of auxiliary lemmas relating to a field often called “geometric al- 
i getra. 1 ' 1 These contain the proofs of theorems related to the techniques of the 
i ^metric application of areas which we discussed in connection with the cube 

* duplication by Menaechmus. 

I fo**Ihc sophistication of its logical structure and the intricacy of some 

V Of its constructions, especially with respect to the irrationals and the regular 

* ^ Elements is predominantly a treatise of an introductory sort, as its 

\ * ,l c im P ,ics - The researches assembled in it were initiated decades before Euclid 

y Hippocrates, Theacletus, and Eudoxus and advanced by their successors, 
v even the organization into treatises comparable to the Elements had been pursued 
y several scholars in the middle and latter parts of the 4th century. 4 Thus, not 
£ * s onc P ut 1° identify any specific theorem in the Elements as the 

| d,scovef y of Euclid, but even his responsibility for the form of the proofs there 
P rc$ ^ n, ^d a nd for the selection and ordering of the theorems in the separate books 

* !lf UllC * n C,car, y< l * ie “Pythagorean theorem’* was familiar a millennium 

I ore Euclid among the mathematicians in ancient Mesopotamia, for instance; 
t Pr °° r in *’ 4? * Which Prodl,s ascribes to Euclid himself, is patently a 

(v* T .* ca ** on lhat in VI, 31, and the result in this latter form can hardly be 
? to Hippocrates, for it is fundamental for his quadrature of the lunulcs.* 

j rvrr* ^ uc ^*^ ean Proportion theory in Book V without doubt incorporates 

• jn i ications on the form of the ihcory initiated by Eudoxus; but several passages 

; \° m ^ sl °tlc’s discussions of the continuum make clear that such changes were 

t a rcacJ y wcil un(] cr way tlircc or four decades before Euclid.* We may say the 

t Same for ,he ,heor y of irrationals in Book X. or even dial Euclidean principle 

P ar excellence, the postulate ol parallels. 7 The key conceptual insights as well 
■ as * c Major portion, if not the whole, of die work of securing them through 

I' ™ al P r °of had already been achieved by Euclid's predecessors. 

• . US ’ a PPropriatc measure of the geometric researches conducted by 

p X UC ld and his contemporaries is to l»c sought not in the FAcments, but in the 
< - and Forismx and (unit's, conceived for abetting the solution of 

| ® . c ! nc problems. But before describing these works, let us turn to the cx- 

mination of a striking problem-solving effort representative of the new dcvel- 
pments very near Euclid's time. 

A LOCUS PROBLEM IN THE ARISTOTELIAN CORPUS 

An odd stroke of good fortune has preserved for us a line example of the synthesis 

... oc ' ls problem from late in the 'Ith century: it is embedded as a lemma 

In • 10 !!! d,scussion °* ,,1C s, 'apc or the rainbow in the Aristotelian Meteoro- 
logica (III, and rcads (hus; 

^ C ' hc P° ,n,s K - H have been given, so also would the (line) Kll be given, and 
so 'he (line) MM (is given), so that also the ratio or MM to MK. Thus M touches 
a given circumference." 



0 B Z 

j I-1-1-» 

Figure I 

i 

t 

! In the terminology of the "givens” (Jala) standard in the analytic literature, 

• this expresses the result that the locus of points whose distances from two given 

i points arc in a given ratio (other than unity) is a given circle. This passage is 

! unique in the Aristotelian corpus in that it provides a full geometric demonstra¬ 

tion, and all the more remarkable in that the manner of proof is identical to that 
employed a century later by Apollonius in his second book On Plane Loci, by 
j virtue of which the construction now goes under the name of the "Apollonian 

circles."* Its proof should thus inform us of the technique adopted at an early 
phase of the problem-solving tindilion. 

The writer introduces a line segment and divides it into parts D and B such 
that D : B = MU : MK, the given ratio (see Fig. I)." 1 He then requires that 
there be produced (prospeporistho) an addition Z to B such that I): B = B t- 
Z : I.), and that accordingly. Kll be extended to P such that Z : KH = B : KP. 
He joins MP and claims that "P will be the center (polos) of the circle toward 
which the lines from K fall," that is, the broken lines KMH satisfying the 
condition of the locus. Ixt the line PR he taken such that Z : B : D = KH : 
KP : PR. so that PR : KP = PH : PR. This means that the triangles PKR and 
PRH are similar, for they have the angle KPR in common. Thus, PR : KP * 
HP : PR = 1) : B . the given ratio. Hence, on the onc hand, R is a point on 
the locus, while on the other, PR is the mean proportional of KP, PH whose 
magnitudes are given (that is, independent of the choice of R) so that PR is also 
given in magnitude. Hence. PR = PM and the locus required is the circle of 
center P and raditis PM. 

litis passage is studded with interpretive puzzles. First, there is the question 
of the relation between the Aristotelian and Apollonian versions of the proof. 
The constructions ate evidently the same, hut there arc differences in the dem¬ 
onstrations. The Aristotelian form adopts an indirect proof, in which the auxiliary 
line PR is hypothesized to he greater or less than PM, and it is deduced that 
both HR : RK and HM : MK equal (he given ratio l) : B. But why this should 
be an apparent contradiction is not quite clear. Perhaps the writer has assumed 
the special context of his construction, where the trial point R is to be taken 




J04 Ancient liailmon »»l ticoiiiciiii* Piuhlcms 


llic Generation ofl buclid H>.S 


i- 

h 


along the circumference of the circle HAM; in this case it is indeed true that for 
PR less than PM, the ratio HR ; RK must be greater than HM : MK, since HR 
will be greater than HM and RK will be less than MK. Otherwise, he seems to 
beg the question in supposing as obvious that if R docs not lie on the arc Mn 
the ratio HR ; RK must be different from the given ratio. 11 In the Apollonian 
version the argument is managed in direct fashion: the magnitude PM is not 
introduced, so that the deduction that points on the circle of radius PR satisfy 
the condition that HR : RK equals the given ratio D : B is precisely what has 
been sought, that this circle is the solving locus. In this way, the Apollonian 
version has considerably improved the logical structure of the proof. On the 
other hand, the Aristotelian version concludes at once that PR : KP = HP : PR 
from the similarity of the triangles PKR, PRII. By contrast, the Apollonian 
version, despite awareness of the same pair of similar triangles, launches into 
an incredible argument of a dozen steps to deduce the same proportionality of 
lines. I find it hard to conceive how Apollonius could have missed the trans¬ 
parency of this step, and thus I Would assign the longer detour to a later editor, 
possibly Eutocius, momentarily puzzled by a step assumed as obvious in Apol¬ 
lonius' proof. 1 * 

A second puzzle is that this construction seems to be more complicated than 
it need be. Simson has proposed a very neat alternative form which develops 
from the condition of the locus in a more natural manner. 1 ' Consider what an 
analysis of the problem might be: given the points H. K, wc choose any point 
M such that HM : MK equals the given ratio. (Sec Fig 2.) Divide HK at L so 
that the ratio of the segments IIL : LK equals the same; then by Elements VI, 
3, the line ML bisects the angle HMK. Now, extend HM beyond M to S and 
draw MT to bisect the angle SMK; one then has TH : TK = HM ; MK. M Thus, 
points L and T. as well as M, lie on the locus. Since ML and MT arc the bisectors 
of supplementary angles, angle LMT is a right angle. We may thus draw the 
circle through L, M, T of diameter LT. As ihe points L, T arc given via the 
given points and ratio which define the Incus, and as L, T in their turn specify 
the circle of diameter LT. it follows that all points M of the locus lie on that 
circle. Thus, to construct the locus, one need merely determine the points L, T 
and draw the circle. This form of the solution is manifestly simpler than the 

S 



Figure 2 


I 

I 


I 

J 

I 


( 


l 


M 



Z 

ti 

D 


Figure 3 


Aristotelian/Apollonian form. The contrast is sharpened when one observes that 
the points L. T do not enter into the latter form at all. Perhaps the originator of 
the ancient version did not have at hand the theorem on the bisector of the 
external angle, so critical for Simson's alternative. But one gains nothing in 
attempting to second-guess the ancient writer. What is striking is that Apollonius 
has sustained this relatively cumbersome approach to the problem. Not only is 
his dc|>cndcncc on a model essentially identical to the Aristotelian version thus 
affirmed, but also his treatment of earlier results is revealed. Apparently, his 
aim in the Plane hn i was not to redo the older solutions, but rather to collect 
these and use them as preliminary to the presentation of his own findings. 

This also shows (hat the analysis implicit in the ancient construction of this 
locus could not have proceeded along the lines suggested by Simson. 15 The 
implied analysis would appear to have taken the following form: given the points 
II, K, wc assume the point M on the locus, so that IIM : MK equals the given 
ratio. (Sec Fig. 3| If we now extend HK and draw PM such that angle PMK 
equals angle KIIM, then the triangles PMII, PKM will be similar, since they 
share the angle at P, while the angles at M and at H are equal. Thus, HM : MK 
- Pll : PM = PM : PK, so that each equals the given ratio, and PM is the 
mean proportional of Pll, PK. Consider lines Z, B, D such that Z : B : D - 
HK : KP : PM, Since I) : B is a given ratio, and D is the mean proportional 
between B and B t Z , it follows that Z is given from D and B. Since HK is 
also given, KP and PM are given in magnitude, for they are each in a given 
ratio to HK. Thus, the locus is seen to he the circle of center P and radius equal 
to PM. The construction and synthesis answering to this analysis are evidently 
those given in the Aristotelian/Apollonian solution; and in the presence of such 
an analysis, that solution loses all of the complexity which marks the synthesis 
when it stands by itself. Indeed, the construction is seen to hinge on a single 
insight, a rather clever one, of introducing the angle at M equal to that at H. In 
this way, Apollonius' decision to retain the older form, rather than search for a 
handier alternative, becomes easy to understand. 1 * 


i 




106 Ancient Tradition of Geometric Problems 


The Generation ol IuicIkI U)7 



Our third puzzle is, why docs this construction appear in the Meteorologica 

at all? For his explanation of the circular shape of the rainbow, the Aristotelian 

writer requires that the point M, marking the elevation of the bow above the 

horizon, be determinate. Upon rotation of the figure about the axis HKP, the 

point M will trace a circle, of which the half above the horizon defines the 

position of the whole bow (see Fig. 4). 12 tils stipulation for M is that it be the 

point on the given circle HAM for which the distances HM, MK are in a given 

ratio. This reasonably opaque assumption is not motivated through any physical 

considerations, such as the nature of the optical rays and their reflection; nor is 

any attempt made actually to specify the value of this ratio. Thus, we arc hardly 

any better off than if he had simply stipulated that the angle PKM be a constant. 

*5 Furthermore, he applies the condition in an unnecessarily complicated way. For, 

l as we have seen, if M is taken on the circumference such that HM : MK equals 

> the given ratio, then for any point lower than M, the corresponding ratio will 

t be greater than the given, while for any point higher than M, the ratio will be 

{: less. This is the view adopted by the later Aristotelian commentators, Alexander 

and Olympiodorus, who paraphrase in minute detail the entire construction and 

v proof of the locus problem, although they have utterly failed to perceive its 

.■ significance. 1 * These considerations make plain the derivative character of this 

f; context of the locus problem. The author of the Meteorologica has recognized 

j* that this construction offers one method for fixing the elevation of the rainbow. 

Ij. But the absence of the analysis leaves the construction without clear geometric 

j; motivation, while the stipulation of lines in given ratio stands without physical 

:> motivation. Moreover, the unnatural manner of deploying this condition within 

; the special context of the passage—that is, presenting the general solution of 

!;. the locus, rather than merely arguing the uniqueness of the point on the given 

circle satisfying the condition on ratios—seems to indicate that the writer was 

\i sufficiently impressed by this particular result to create a place for it within his 

V 

; • 


meteorological discussion. That is, it was taken to be a significant discovery, 
and that, we should suppose, was by reason of its being recent and representative 
of advanced study in geometry at the time. ,,, 

In speaking of the “Aristotelian writer/' 1 betray a reluctance to lake Aristotle 
as the author of this solution of the locus problem. It ranks among the finest 
samples of geometry we know of from the pre-Euclidean period, and nothing 
in the Aristotelian corpus prepares us for the notion that Aristotle himself was 
capable of such a technical feat. 241 Hie passage is singular on several accounts: 
only here and in a passage two chapters earlier in the same book of the Meteor • 
ologica docs one come upon the presentation with full proof of a geometric 
proposition. The nearest parallels are a set of theorems on proportions involving 
infinite magnitudes given at several places in the Physics and De Caelo and 
effected with no remarkable skill or insight. 21 Otherwise, Aristotle’s mathe¬ 
matical passages either merely allude to aspects of proofs or take up the discussion 
of mctamathcmatical issues. These arc valuable as reflections of mathematical 
work of the period, but they provide no evidence of Aristotle’s active participation 
in that work. Furthermore, one may well entertain suspicions about other features 
of the passage in the Meteorological the optical ray is conceived as emanating 
from the eye and outward to the object seen, here front K to II via reflection at 
M; this conforms with standard usage in ancient optical studies, but directly 
contradicts the theory of vision Aristotle favors against this view in l)c Aninut 
(II, 7), in which seeing is an effect of the transparent medium acting on the eye, 
so that the account of vision harmonizes with the accounts of the other senses. 
Why should Aristotle not have adopted the same view in the Meteorologica*} It 
is geometrically indifferent, and, indeed, optical writers sometimes do adopt it 
in a context, as here, where the object seen is a luminous body like the sun or 
a star. 21 This discrepancy has long been noted by commentators, both ancient 
and modern; but it seems not to have provoked the uneasiness one should have 
expected. 2 ' Again, the analysis of the rainbow introduces the bizarre device of 
placing the sun (II) and the cloud producing the rainbow (M) at equal distances 
from the observer (K). This fiction, sometimes called the “meteorological hemi¬ 
sphere,” 24 is patently absurd from the cosmological standpoint, as Aristotle must 
have known, in view of his familiarity with the Eudoxcan astronomy (in 
Metaphysics XII, H) and his consistent distinction between the sublunary sphere 
of the elements and the higher domain of the heavenly bodies. Why should 
Aristotle here so casually evade the plain principles of cosmology? If one sup¬ 
poses that he merely adopts a standard meteorological convention, just as we 
saw before he followed the optical convention in the conception of the visual 
ray, then one must admit that Aristotle’s thought has had no impact on the 
formation of those conventions among the mathematical scientists of his day. 
Yet the adoption of this convention seems odder still in that it provides no 
particular advantage for the account of the rainbow. Conceiving the sun in its 
true position merely renders the ray HM effectively parallel to the horizon, 25 
and so would require an alternative condition for determining the altitude of M, 
for the condition on the ratio of the lines HM, MK becomes meaningless. This 






108 Ancient Tradition of Geometric Problems 


The Generation of Kuclid 100 


seems. Ihcn. to be another indication that the account of the rainhow has been 
fashioned to serve as the vehicle for this problem, rather than to provide the 
context for its original statement and solution. 

If these considerations do not quite rule out (he authenticity of Meteorologica 
Book HI. they surely deserve more serious attention than Aristotle scholars tend 
to afford them. At the least, we recognize that the geometric locus problem in 
the passage on (he rainbow is an importation from geometric practice, not Ar¬ 
istotle's own discovery. But the suggestion of its nonauthcnticity at once raises 
the problem of dating. It has already been observed that the Meteorologica 
appears to be a composite document, different sections reflecting times ranging 
from the mid-350's to the late 340\s. ,ft Conceivably, then, some of its parts 
might be later still, and some might even be the work of disciples after Aristotle's 
death in 322. If so. how much later? If the passage on the locus problem derives 
from the 3rd century, near or even after the time of Apollonius, then its signif¬ 
icance for the history of ancient geometry diminishes almost to nil. 

In this instance we possess a valuable index of early provenance: the frequent 
appearances of archaic terminology in expressions like “the line on which (arc) 
AB” (he gramme eph* lici AB ), where standard usage alter Kuclid would read 
“the right line AB" (he euthvia AB). ?1 We have seen that the older form is 
characteristic of usage in the Hudemus fragment on Hippocrates cited by Sim¬ 
plicius, and an analogous form for logical terms is prevalent in Aristotle’s Prior 
Analytics. By contrast, the shorter form is exclusive in Kuclid. Archimedes, and 
Apollonius and the writers who followed them.*" One must suppose that the 
author of the solution of the locus problem was in touch with the regular tradition 
of geometry of his time, so that if that happened to he much alter the beginning 
of the 3rd century, the use of such nonstandard terminology would be difficult 
to explain. 

Thus, I view the account of the rainbow in the Meteorological and the version 
of the locus problem to which it has been specially accommodated, as a product 
of the late 4lh century, written more likely by a disciple of Aristotle than by the 
\- philosopher himself. The solution of that problem is not original in this context, 

£ as one sees from the absence of the analysis which would motivate the construe- 

5 ' don, but more clearly, from the partial garbling of the proof, needlessly cncum- 
bered by an indirect format of demonstration / 1 What, then, was the nature of 
I the author's source and of the geometric activity which produced it? We have 
s little means for determining this. But certainly when Apollonius reproduced this 
: construction in his Plane Loci a century later, he was drawing not from the 

\ Meteorological hut from its mathematical sources. Ht The Meteorologica passage 

' Ihus reveals to us the manner in which this result was received by a mathemat- 

; ically oriented writer, late in the 4th century, and so indicates both the topicality 

and the significance of its discovery in the field of research then current. 

EUCLID’S ANALYTIC WORKS 

The problem from the Meteorologica manifests several features standard in the 
classical form of problem solving. It is a problem of Incus, that is, the construction 


of the set of points (typically a line or curve) satisfying a stated condition. But 
here it is actually expressed not as a problem, but as a theorem about “givens": 
that if two points arc given and a ratio is given* then the circle is given which 
is such that the lines drawn from each of its points to the given points arc in the 
given ratio. That Kuclid devoted an entire treatise, the Data (“givens"), to 
theorems framed in this manner reveals that the form was important within 
geometric studies of the time. We recall, furthermore, that the Aristotelian 
problem assumes an auxiliary construction: that the line Z may be produced 
[prospeporistho) such that for given lines B, I), I) : B “ B H* Z : 13. Here too 
Kuclid composed a massive treatise of Pori sms* now lost, so called perhaps only 
for their role as auxiliaries to other constructions, although the stronger sense 
of “production," that is. of terms of a stipulated description, might also apply. u 
Most important, we have seen that the Aristotelian problem, while presented in 
strictly synthetic lashion. is best umlcrsttMul as framed in answer to a preliminary 
analysis. Cleatly. this analytic method had attained prominence by Kuclid’s time, 
for Pappus includes these and other treatises by Kuclid among the corpus of 
works, the Topos analvomcnos (frequently translated as “Treasury of Analy* 
sis"), designed in the interests of analytic investigations.'’ 

The Data is a complement to the Elements* recast in a form more serviceable 
for the analysis of problems." As in the Aristotelian example, each of its theorems 
demonstrates that a stated term will he given on the assumption (hat certain other 
terms are given. The subject matter overlaps that of the Elements* dealing with 
ratios and with configurations of lines and of plane figures, both rectilinear and 
circular. Indeed, only in rare instances docs the Data present a result without a 
parallel in the Elements. u The proofs follow in a deductive sequence, so that 
one might refer hack to previously proved theorems; in the Aristotelian prohlcm, 
for instance, one assumes that the ratio ol given terms is given, or that a term 
in given ratio to a given term is itself given—such propositions appear in the 
Data (Prop. 1.2) and must have been available to the Aristotelian writer in a 
comparable hunt. Again, some proofs may assume icsults proved in the Ele¬ 
ments* for instance, properties of similar triangles are needed in the Aristotelian 
problem. In view of this, one might suppose the Data was effectively supcrlkious 
in light of the Elements* 

But the analysis we sketched for (he locus problem in the preceding part 
reveals the special utility of the “data" format for this type of investigation. 
There, the analysis started from the supposition that the construction has been 
done, namely, that the point M is such (hat HM : MK equals the given ratio. 
The construction of P then led to a set of ratios linking the lines HK, KP, PM. 
We next assumed an arbitrary line D as givenx hence B, the line in given ratio 
to D, is also given (so that D:B « PM: KP). Furthermore, if Z is the line 
such that I) is mean proportional between B and B + Z, then it mo is given , 
mul it has to HK the same ratio that B, I) have to KP, PM, respectively. But 
HK is given (for the points II, K are given as basis of the locus to be constructed), 
so that the ratio HK : Z is given ; and B, 13 arc given* so that KP, PM arc given 
(the latter in magnitude only), whence the circle of center P and radius PM is 
given. Thus, the locus of M is given. 



I til' v tcuvi.»ll«*i| ill iakiiu 


i l l 


fc: 


This ends (he analysis and serves as the basis for the synthetic construction 
of the solution, as we meet it in the Aristotelian and Apollonian texts. One 
proceeds generally in the reverse order: starting from the given points H, K and 
the given ratio, one introduces auxiliary lines D, D in that ratio and produces 
the line Z» as described in the analysis; from this the lines KP, PM arc determined 
and the circle is drawn. The proof consists of verifying that HM : MK equals 
the given ratio. In (his instance, each step of the analysis was simply convertible. 
But il sometimes happens that the reversal of the logical order is possible only 
when certain additional conditions arc satisfied; the determination of the appro¬ 
priate conditions would then require a separate treatment, called the "diorix- 
mos/' in effect specifying when the problem as stated is in fact consiructiblc. ' 6 
The investigation of diorisms seems to have been advanced already by the middle 
of the 4th century, so that the procedure of analysis and synthesis was familiar 
as an instrument for the solution of problems several decades before Euclid.' 7 

Referring back to our sample analysis, one observes how the concluding 
section is shaped by a scries of "givens," each determined from preceding ones, 
until one has arrived at (he desideratum of the construction as given. The adoption 
of this terminology is not merely a formalism; it serves a critical purpose by 
keeping separate two sets of terms having quite different logical status. Some 
terms arc known only conditionally, in that they follow upon the hypothesis of 
the analysis that the construction has been effected; hut other terms arc known 
by virtue of the definition of the construction which one is to produce. The latter 
are the "givens." When these (wo sequences of terms arrive at a common term, 
one may consider the analysis to be complete. Without a terminological ploy 
like that of the "givens." the logical status of these sequences would soon 
become hopelessly confused. Thus, far from being superfluous, Euclid’s Data 
accords formal recognition to the advanced stage of analytic researches at his 
time. 

The Meteoralogica passage already employs the form of sequential “givens," 
so that Euclid cannot be taken as having originated this style, although it is 
possible that his Data represents the first major effort to organize the materials 
of elementary geometry in this form. Indeed, some pans of the Data are modeled 
after their analogues in the Elements . indicating that these parts arc not likely 
to have had an independent provenance. 1 * We lack earlier instances of the se¬ 
quential usage, doubtless owing to the general loss, hut for fragments, of doc¬ 
umentation from the pre-Euclidcan period, so that one cannot specify the origin 
of this usage. But for the simple notion of the "given" in geometry, one has 
evidence from early in the 4th century in the "hypothesis" passage from the 
Mena (86e-87b), where the project is to apply a certain area "along the given 
line of this circle here..The phrasing is somewhat loose, as the dialogue 
context would recommend, hut the formal basis can be seen: the conditions to 
be satisfied in the construction of a problem are framed in terms of certain 
"given* 1 magnitudes. This is invariably the form in the problems in Euclid's 
Elements" It is perhaps best to view this as a natural development, formalizing 
usage in common language. For one comes upon comparable expressions in 


dialectical passages, for instance, without geometric connotations. Aristotle de¬ 
scribes the good legislator as "capable ol taking the given constitution into 
account," that is, the one actually existing, rather than some hypothetical ideal 
(Politics IV. I, 1288 h 28); or. again, he dismisses the notion that an eternal 
thing might he subject to generation or passing away, as this "conflicts with 
one of the givens/' that is, the generally accepted principles of natural philos¬ 
ophy . m In view of such parallels, however, one should not rule out the possibility 
that geometry and dialectic interacted in the development of this usage, at a time 
when both fields were seeking to formalize the elements of research. 

A further contribution by Euclid to the study of advanced problems lies in 
the field of the conic sections. In the production of bis treatise On the Conics, 
Euclid was able to draw on the efforts of an earlier contemporary. named Ar- 
istaeus "the elder" by Pappus, for he is said also to have written a treatise on 
this subject. 41 As neither work survives, doubtless owing to their having been 
superseded by Apollonius* far more extensive treatment about a century later, 
our knowledge of their nature is indirect and severely limited. One must suppose 
that whenever Archimedes assumes without proof a theorem on conics, "for 
that has been demonstrated in the Conic Elements/* he refers to the work either 
of Aristaeus or of Euclid. 4 * From Apollonius himself and commentary by Pappus 
and Eulocius. wc learn that most of the materials on the general properties of 
the curves in Apollonius* Books I and II were l ami bar within the older theory, 
and that his third book extended a body of theorems useful for the solution of 
locus problems initiated by Euclid. As for the fourth book, investigating the 
manner of intersections of conics. Apollonius assigns its origins to Conon of 
Samos, a geometer in the generation just after Euclid and an important influence 
on Archimedes* early work. The remaining four books on normals, similar 
conics, and conjugate diameters were the product of Apollonius* own more 
advanced researches. 41 Thus* die theory at Euclid’s lime may he viewed as 
consisting essentially of the materials til the first two books and sonic portions 
of flic third. 44 

As we have seen, the standard view that the theory of the conics was initialed 
and strongly advanced by Menaechnuis around the middle of the 4th century is 
open to serious question. Mis treatment of the cubc-duplication problem indicates 
an interest in the family of curves specified via second-order relations in ac¬ 
cordance with the techniques of the application of areas, and he and his followers 
surely were able to discover properties of such curves and use them in the solution 
of other problems. 4 * But the original context of the discovery that such curves 
resulted from the sectioning of cones is a matter of conjecture. O. Neugcbaucr, 
for instance, has suggested the study of the shadows cast in sundials as a possible 
context. 4 * Alternatively, the field of optics seems a likely prospect. For instance, 
Euclid demonstrates that "the wheels of chariots appear sometimes circular, 
sometimes drawn in (/nirespasmenoi)" (Optics, Prop. 36), and Pappus includes 
among his lemmas to the Optics the construction of the ellipse whose appearance 
the circle assumes in the latter case. 4 ' Moreover, the earliest extant references 
to the conic section have to do with the same situation: for instance, that the 


r *«* 





\ gibbous moon looks like a "‘shickr* {ihyreox ) or lliat a circle viewed ohlit|iicly 
l lakes the shape of “(lie section of an acute-angled cone.” 1 " 
f Even if my view of (he earliesi phase of die study is correct. we should slid 

; recognize that die Greek geometers could noi long have icniained content with 
l <he type of poiutwisc constructions we have proposed. Curves arose as ahstrac- 

• dons of motions (c.g., circles and spirals) or through the intersection of solids 
(e.g., the curve of Archytas). The sort of algebraic relation by which one defines 

.. a curve in the Cartesian manner was admitted by the Greek as a fundamental 
■ properly (symptdma) of the curve, but not as the basis of its conceptualization. 4 * 

| One can imagine how the search for a natural mode of generation might have 

i progressed from the pointwisc construction to the sectioning of cones. I'm in- 
} stance, the curve introduced via the relation Y*’ •= AX might entail for its point- 

* w * se construction a diagram consisting of a system of circles sharing a vertex 
and having their diameter lying along the same line |Fig. 5; ef. Cli. 3, Fig. 
12 (c)); for each circle the chord is drawn perpendicular to that diameter, as to 


divide the diameter into segments equal to A and X. so that the chord itscll will 
be twice Y. v ’ One might tlicn see. if tl»e lines X arc drawn in elevation at right 
angles to the plane, that the diagram becomes the plane projection of a cone, 
one of its generators being the line perpendicular to the plane and passing through 
the common vertex of the circles, and the curve being formed as the cone s 
intersection with a plane parallel to that generator. 

Under this view, the study of these curves, as dclincd via the application of 
areas, had progressed significantly before it was discovered that the curves formed 
as sections of the cone satisfy the same second-order relations. The work of 
producing a full Hedged treatise on the conic curves, like those by Aristacus and 
Euclid, could proceed quite rapidly, for most of the properties of these curves 
arc deduced from their xymptdmata. the second-order relations, rather than from 
their generation as sections. It is important to realize that the role of the latter 
aspect of the curves was to provide a natural conceptual view of their generation. 

It did not serve for their practical construction, for later writers found pointwisc 
procedures far more convenient than any solid construction. 1 On the other hand, 
it was largely suiierfluous for the work ol proving theorems, at least in the more 
advanced stage of the theory at the hands of Apollonius. Realizing this helps to 
explain another rather puzzling feature of the early theory of the conics. We 
learn from Pappus ami Kntocius that before Apollonius the conics were specified 
such that the sectioning plane was taken perpendicular to the side of the cone.’' 
Thus, parabolas were formed only via the perpendicular sectioning of the right 
cone with a right angle at its vertex, and were accordingly named "sections of 
the right-angled cone.” Similarly, ellipses and hyperholns were formed, re¬ 
spectively. from cones with acute and obtuse angles at the vertices. The report 
of the commentators is continued by usage in Archimedes and Diodes, and the 
older names persist in some accounts well* after Apollonius. M Yet why this 
restriction? It seems to lielic the fact that specimens of all three types can be 
formed as sections from any cone • -a consideration which surely must have led 
A|Millouius to remove this restriction tin sections, although the nature of oblique 
sections was already well appreciated by Euclid and Archimedes before him. 
Odder still, it prohibits the formation of a circle as a "conic section.” Clearly, 
then, the retention of this mode for so long was due to its introduction as a 
deliberately artificial aspect of the theory; that is. it reflected not the primitivencss. 
hut the sophistication of that theory around the time of Euclid. 

Our account of the initial phase of these studies revealed that the second- 
order curves at first lacked a natural mode for their generation. The sectioning 
of the cone could provide a suitable mode, and indeed under the restricted manner 
of sectioning there will lie a unique section corresponding to each second-order 
curve . u To sec this, first consider the parabola: in the application mode it is 
specified via a single parameter, the line segment over which the application is 
performed (line A in the relation Y J = AX). Correspondingly, when it is formed 
from the perpendicular section of a right-angled cone, a single term, the distance 
D of the cutting plane from the vertex of the cone, will designate the curve. 
|See Fig. 6(a). | It follows that A = 21). Since the distance along the diameter 


of the section from its vertex to where it meets the axis of the cone also equals 
D, we obtain a straightforward sense for Archimedes' expression of the parameter 
as “double the line up to the axis."" Turning now to the hyperbola and the 
ellipse, we may specify each form via two lines such that an area similar to the 
rectangle defined hy the two lines is applied to one of those lines as to exceed 
or fall short by a square (i.e., the lines A, C in the relations 

- YY = AX + X 1 
r 

where the positive sign corresponds to a hyperbola, the negative sign to an 
ellipse). We have two important indices supporting this view of the initial forms 
of the curves. First, it leads immediately to the form implied as the standard in 
Archimedes' theorems, namely as Y*: X (A ± X) « C : A.' 6 Second, Pappus' 
statement of the definitions explicitly figures the excess or defect as a square. 
But in Apollonius* definitions there is a subtle difference; for the “lalus rectum" 
C and “diameter" A give rise to the relation 

Y J = CX i -X • X 

A 

so that the excess or defect is similar to the figure defined by the lines C, A. 
Pappus has not erred, as some have charged; he has merely followed a source 
which conforms with the prc-Apollonian designations. 51 Like the parabola, each 
hyperbola and ellipse may be associated uniquely with a perpendicular section 
of a cone |sce Fig. 6(h) and 6(0), where the distance I) between the vertex of 
the section and the vertex of the cone and the angle 0 at the vertex of the cone 


are determined from the lines A, C via the relations C = 2D tan {'A) 0 and 
A = I) (an ft (lor Ihe ellipse, 0 acute) or A = D tamp (for ihe hyperbola.* ft 
obtuse and q> its supplement).'* Note that just as for the parabola, the lalus rectum 
may be styled as “double the line up to the axis." 








.Uu m mi I t.iiKiiiiii til v UK I minii(i:> 


This effort lo relate the second-order relations to the sections of the cone 
produced constructions to form an appropriate conceptual background for their 
study. But it would not much affect the content or order of the theory, or its 
primary purpose, to serve as an instrument for the investigation of problems. 
From the start, it was in their form as area relations that the curves had their 
application, and this remained so throughout the later development of the theory. 
We have seen how the cube duplication figured prominently in the initial phases 
of this work, and we shall see in the angle triscclion another significant example 
of its use. Apollonius informs us that Euclid made a start in another area where 
the conics are indispensable: the study of the problem of the locus of three and 
four lines.** In this, one seeks a construction of the set of points such that given 
three (or four) lines, the distances A, B, C (and D) of each point lo the respective 
given lines satisfy the condition that the ratio A ■ B : C 2 (or A ■ B : C * D) is 
given. In general, the locus is a conic section. Theorems from Apollonius* third 
book (Prop. 16-22), on the ratios of segments of tangents and of intersecting 
chords, have (heir application in the solution of this problem, as Apollonius 
observes, noting in strong terms the inadequacy of Huclid*s version: “lie did 
not effect the synthesis |of this locus| save for the chance part of it and that not 
successfully,** for only with the aid of Apollonius’ own newly discovered theo¬ 
rems could that be done. Zeuthen has suggested that Euclid would have been 
hindered in this effort through his failure to conceive of the two branches of the 
hyperbola as constituting a single curve.* 1 But it seems that Apollonius levels a 
more serious charge than this would indicate. Unfortunately, understanding this 
issue depends heavily on reconstructions, for neither the Euclidean nor the Apol¬ 
lonian investigation of this locus survives. But it can be Imped that insight into 
the former may follow from consideration of another of the Euclidean works 
Pappus includes in the analytic corpus, the Porisms. 

The three books of Euclid’s Porisms, according to Pappus, formed a massive 
treatise of 171 theorems falling into 29 classes, yet without claim of exhausting 
the subject matter.* 1 Pappus gives a paraphrase collecting into a single proposition 
several of Euclid’s; the first part, incorporating ten prn|tosiiioiis of the first 
species, is this: 

if three points on one line of the hyption or parhyptitm figure arc given, and the 

others save for one lie on a line given in position, then that also lies on a line 

given in position. 

That is, with respect to the two configurations, the “supine” figure |Fig. 7(a)| 
and the “hypcrsupinc” figures |Eigs. 7(b) and 7(c)|, if points A, B, F arc given 
and points C, D lie on given lines, then E also lies on a given line.* 2 One 
recognizes the connection with those studies now included within the field of 
projective geometry and which received their modem treatment through re¬ 
searches initiated in the 17th century by Dcsargucs and Pascal. 6 ' For instance, 
the converse of the lemma just cited from Pappus provides a proof of the theorem 
of Oesargucs on two projeciivcly related triangles: (hat if the lines joining cor¬ 
responding vertices of the triangles CEO, C'E'D' meet at a point, then the points 
of intersection A, B, F of corresponding sides arc collincar |scc Fig. 7(d)|. 64 A 



perusal of the W lemmas which Pappus presents on the Porisms confirms this 
sense of the affinities between the subject matter of the Porisms and that of the 
more modem field of projective geometry. In addition to the connection with 
Dcsargucs* theorem just cited, one finds configurations relating lo the complete 
quadrilateral, the preservation of cross ratio through projective transformations, 
and results on involutions. Most noteworthy is the famous theorem of Pappus 
on the hexagon inscribed hetween two lines: 

if AH, CiD are parallel lines and there fall on them certain lines AD. A/.. BG, BZ 

and El>. EG arc joined, then a straight line arises through H. M, K. M 






•? ^.T r . -yenpiwr w** 


.•IIM'I'II Ml MI.IMIHIIK I IMllU lll\ 



|Scc Fig. 8(a). | With such extensive information to guide them, several scholars 
have attempted reconstructions of this lost work. The most ambitious effort is 
that of Chasles, who perceived a clear link between the ancient study and his 
own researches in projective geometry. 6 * Working through the many cases sug¬ 
gested by configurations such as these. Chaslcs compiled a series of over 200 
propositions whose demonstrations could be managed within the technical do¬ 
main available to Pappus. However impressive this project might seem, others 
appear to have been disappointed. For surely Pappus' lemmas arc but a mere 
shadow of the scope of Euclid’s work itself. 67 Thus. Zcuthen refers to the more 
general context of these studies in the modern theory— for instance, the theorem 
of Pappus is a special case of Pascal’s theorem on hexagons inscribed in conics— 
and so proposes that Euclid was moving toward a projective theory of the conic 
sections. 6 " 

It would appear, however, to be a questionable procedure on our part to 
extrapolate in this way and insist that because the potential for such a theory is 
foreshadowed in these lemmas, the work to which th^y relate, Euclid's Porismx , 
must actually have advanced such a theory. It is an interpretive issue of no small 
subtlety to determine in what sense and to what degree Euclid and later geometers 
developed a certain body of geometric materials which have been subsumed in 
more recent times within the field of projective geometry. This caution is es¬ 
pecially well advised because the ancient manner of effecting these results entirely 
lacks the conceptions and methods characteristic of the modem field. Consider, 
for instance. Pappus' treatment of the theorem on the inscribed hexagon. He 
establishes this in two cases, as the hounding lines are parallel (l«cmma 12 ) and 
as they intersect (Lemma 13) 6 *; in this he already betrays the absence of an 
essential feature of projective methods: the equivalence of parallel and nonparallcl 
configurations through the introduction of points and lines at infinity. In the 
nonparallel case, with reference to the lines CIHTE and NGZD, Pappus has, 
from a prior lemma ( 3 ), CJE • HT : GH • TE « GN * ZD : ND * GZ; similarly, 
with reference to NGZD and DKLE, he has GN • ZD : ND • GZ - DK • EL : 
DE • KL |scc Fig. 8 (b )!." 1 It thus follows that GE • IIT : GH * TE = DK • EL : 


I lie iicm'i.iiuiii Ml l *ik ml 


117 



DE * KL. He can thus conclude from another lemma (10) that IIMK is a straight 
line. Pappus' proof is structurally identical to the standard projective one: the 
perspectivity centered on A projects GIITE onto GZDN, while that centered on 
B projects GZDN onto LKDE; the projection GHTE onto LKDE is the compo¬ 
sition of these two pcrspectivities, and since it has the fixed point E, it is itself 
a perspectivity. The center of projection must be the intersection of GL. TD, 
namely, the point M; hence the line HK passes through M. 7 ' From a comparison 
of these two treatments, one secs that Pappus* Lemma 3 amounts to the specifi¬ 
cation of a projective transformation (here, a perspectivity) via its cross-ratio- 
preserving property; its converse (Lemma 10) plays the role which the modem 
approach casts to the fundamental theorems on the composition and determination 
of prqjcciivitics ami pcrspectivities. 

There emerges from this a tantalizing puzzle: in view of the utter absence of 
projective techniques from the ancient treatment, how did it happen that the 
author of the Porismx came to investigate the same configurations and properties 
which are prominent in the modem theory? Just what was Euclid up to? What 
wc know of this work indicates that it was ancillary to a larger field of research . 72 
Indeed, Pappus says so explicitly: "the Porismx of Euclid are a most ingenious 
set mustered toward the analysis of the more heavy-laden problems," and notes 
their utility for locus problems of the son found in abundance in the topos 
analyomenox. 1 ' Now, as wc have noted, results comparable to some in the 
Porismx have been applied with stunning success toward the study of conics 
since the 17th century. In the light of Euclid's contributions to the early theory 
of conics, one may well conjecture that Euclid compiled the Porixms to support 
a problem-solving activity related to the conics. This view would be a somewhat 
weakened form of that advocated by Zcuthen. Let us then consider one of the 
problems of this soil expressly assigned to Euclid, the locus with respect to three 





E 



A 


120 Ancient Tradition of Geotnetne Problems 

and four lines, to see whether it can provide insight into the nature and the role 
of the Pori sms. 

THE ANALYSIS OF CONIC PROBLEMS: 

SOME RECONSTRUCTIONS 

One may observe that the prc-Apollonian forms for the central conics express 
them as special eases of the three-line locus. In the defining relations Y* : X • 
(PP' t X) - L : PP\ for given lines L, PP', the lengths X and PP' ± X 
represent the distances of each point of the conic from two given parallel lines, 
as measured in the direction of n given transversal line, while Y represents its 
distance from that transversal, as measured in the direction of the parallels (see 
Fig. 9). One thus sees that the ellipse (for PP' - X) and the hyperbola (for 
PP' + X) which are tangent to the parallel lines at P t P' and have PP' as diameter 
and L as latus rectum will satisfy the properly of the three-line locus relative to 
these given lines and the given ratio L : PP # . In effect, then, the three-line locus 
problem seeks to generalize the defining condition of the conics for other con¬ 
figurations of the reference lines. 

Solutions of the genera) problem can be constructed on the basis of two 
propositions from Apollonius’ Conic*: 

III. 16 : if tIP. OO are tangent to a conic amt KK' is a secant line parallel lo OQ 
which meets the conic in R, then OP’ : 0Q ; = PK’ : KR • RK* (see Pig. IU(;»)|. 

III. 17: if RR\ SS' arc secants parallel to OP. UQ. respectively, and meet each 
other in J. then 0P : : OQ' = RJ • JR' : SJ * JS\ (See Pig I0thl|‘ 4 

The latter theorem holds not only when J is an internal point (as shown in Fig. 
10b), but also when the secants intersect externally. Hie theorems arc related 



the Cicnciitli*>n *•! taicltd IJ i 




Figure 10 

in that they each generalize familiar properties of tangents and secants to the 
circle, where OP - OQ. PK J = KR • RK’ and RJ * JR' - SJ - JS' <cf. Euclid's 
Elements III, 35, 3fi). Since Archimedes asserts as “proved in the Conic Ele¬ 
ments" this same property of the intersecting chords in a conic (Conoids atul 
Spheroids . Plop. 3|. M we thus perceive that this property was established within 
the earlier theory of conics. 

The fact that a conic has the property of the three-line locus follows at once 
from III. If*. 1 * for if we draw tangents TP. TP' to the conic, bisect chord PP' 
in V, and join TV meeting the conic in Q, then TV will be a diameter corre¬ 
sponding lo the ordinates parallel to PP' |scc Fig. I l(a)| M Moreover, if OQ is 



Figure 9 


Figure 11 







122 Ancient Tradition til Geometric Problems 


i in* orm i*niifii mi i.iuiiw 



drawn parallel to PP', it will be tangent to the conic at Q. If next we draw lines 
from R parallel to TP, TV, TP' meeting PP', respectively, in S, V\ S', then 
KR = PS, RK' = S'P' and PK : RV' = PT : TV. We thus have 
OP 2 : OQ 2 = PK 2 : KR * RK' - (RV' 2 : PS • S'P') • (PT 2 : TV 2 ), so (hat 
RV' 2 : PS • S'P' has a given value, independent of the choice of point R on the 
conic. As the lengths PS, S'P', RV' are the respective distances of R from the 
lines TP, TP', PP' as measured in the given directions of PP\ TV, one sees 
that R lies on the three-line focus relative to the lines TP, TP'. PP', and the 
given value of the ratio. 

Conversely, to find the curve answering the condition of the three-line locus, 
we note that if Q is the intersection of the curve with TV, then QV 2 : 
PS • S’P' = M 2 : N 2 , a given ratio |scc Fig. I |fb)|. Since PS - S'P' = OQ, 
and OQ : TQ = PV ; TV. one has QV : TQ = <M : N) (PV : TV), a given 
ratio. Wc may thus determine Q on the given line VT by means of this proportion 
and then introduce the conic of vertex Q, diameter TV and tangents TP, TP'. 
That is, when Q bisects VT, the conic is the parabola whose latus rectum L 
satisfies the relation PV 2 = QV • L. Otherwise, wc find Q' via the harmonic 
relation QV : VQ' = QT : TQ'; then QQ' will be the diameter of the conic and 
L its latus rectum, for PV 2 : QV • VQ' = L : QQ'. One notes that this yields 
an ellipse when QV is less than TQ and a hyperbola when it is greater. 

In this way. the solution of the three-line problem, where the solving locus 
is a parabola, an ellipse, or a single branch of a hyperbola, was available to 
anyone knowing the property proved in Conics 111, 16. An aspect of the problem 
not within the range of the theory of conics before Apollonius relates to the 
cases where both branches of the hyperbola arc considered. 7 " We may recognize 
two additional configurations analogous to III, 16: one in which the tangents are 
drawn to P, P' on one branch and R lies on the second: another in which the 
tangents arc drawn to different branches. Apollonius presents the analogues in 
III, 18, 19 from which one can complete the construction of the locus. 



Having established that the conic satisfies the property of the three-line locus, 
one can derive that it satisfies the property of the four-line locus by considering 
the tangents drawn to the four vertices of an inscribed quadrilateral and applying 
the three-line result in succession to each of the four sides |scc Pig. I l(c)|. w 
Rut the conversion to a construction of the four-line locus is not as straightforward 
as that of the three-line locus. Zeulhen has thus proposed a different construction 
developing from the relation of the segments of intersecting chords (III, 
lit the first instance, let the given four lines intersect in the trapezium ABC!), 
sides Al) and RC being parallel; wc wish then to determine the locus of points 
R such that Rl * RK : Al • IB lias a given value, the four segments here being 
the distances from R to the respective sides of ABCI) as measured in the directions 
Al), AB (see Pig. 12). l.et us conceive the conic curve about ABCI) whose 
diameter is the bisector of the parallel sides Al), BC and whose ordinates lie in 
the direction of Al). Then by III. 17, the ratio Rl • IS : Al • IB has a constant 
value for all chords drawn parallel to fixed directions, while RK = IS. Ilcncc, 
this conic will he a solution for the locus of R if it can be contrived that this 
ratio has the value specified in the locus problem when the chords arc parallel 
to AD, AB, respectively. To construct this conic, Zeuthen proposes the intro¬ 
duction of an auxiliary conic which will turn out to be similar to that required. 
Since (he tangents parallel to AD, AB meet in segments such (hat the ratio of 
their squares OP 2 : OQ 2 equals that of the products of the segments of the 
intersecting chords, wc may assign segments 0 'P\ O'Q' respectively parallel to 
AD, AB, such that O'P ' 2 : O'Q ' 2 has the given value, and draw PT’ parallel to 
the bisector of Al), BC. In this way, O'P', O'Q' will he tangents and P'T' the 
direction of the diameter of the auxiliary conic similar to the one sought. Fxtcnd 







)24 Ancient I (minion oi (icniiiiinc Problem* 



O'Q' to meet P'T' at V\ and draw from Q r a line parallel to O'P' meeting P'T' 
at W\ We now produce the point X' on P'T* in harmonic relation with V\ P\ 
W‘; that is, PV' : P'W' = X'V' : X'W\ Thus, P'X' is the diameter of the 
conic (via Conics I, 36) and its latus rectum L' is determined from the relation 
Q'W° : P'W' * W'X' = 1/ : P'X' (via I, 21). To complete the const met ion, 
one must circumscribe about ADO) the conic similar to that in the auxiliary 
figure. For instance, since the bisector K' of P'X' is the center of the auxiliary 
conic, the direction K’Q' is known; (he analogous line in the required figure 
must meet the diameter at the same angle and pass through the midpoint M of 
AB, a chord parallel to the tangent corresponding to UQ. This determines the 
center E of the required conic. If we join E, A, the direction of the corresponding 
line in the auxiliary conic is the same, whence A' is found as the correlate of 
A. We can now inscribe the trapezium A'B'C'D', similar to ABCI), and from 
it determine the points 0, P, y. X which specify the conic. Alternatively, once 
the ratio EQ : EP ol conjugate diameters has been found. Zeuthen proposes an 
application of Conics III, 27 from which the diameter of the required conic can 
be determined, thence the conic itself.” 1 

In Zeuthen's account, not only could Euclid produce a solution of this sort 
for the ease of the four-line locus where the reference figure is a trapezium, but 
also he could extend this to the general quadrilateral. For this Zeuthen uses a 
lemma based on one of the locus propositions Pappus cites from the Ponsms. H} 



Unfortunately. Zeuthen imist assume that Euclid had access to an extension of 
the cited proposition in which the Ittcus of a given point can be taken to be not 
only a line, but also a conic (that is, a curve described by the fourdinc locus). 
But one may also see that Zeuthen's method is unnecessarily complicated; for 
the transition from the trapezium to the general quadrilateral can be effected 
without recourse to the lemma he employs. I.el (he quadrilateral ABCI) be the 
reference figure for a four-line locus, such that distances are to he measured in 
directions parallel to AH. HC (sec Fig. 13). We complete the parallelogram 
AIUT. and extend AF. to meet the solution curve of the locus in G; let AG meet 
Cl) in II and let Al) meet HC m K; and lei the line from G paiallcl to KC meet 
AI) in I. and IK' in M. Since G is on the locus. Gil • GA : GL * GM = r : 
the given ratio; while GM = EC and GA : GL = AE : EK via similar triangles. 
This determines Gil, so that the given figure ABCD determines G on the locus. 
Since the live points A, B. (\ I). G determine the conic uniquely, one is thus 
permitted to solve the locus via the trapezium AllCCi." 1 

As in the ease of the three-line locus, the complete solution of the four-line 
locus would he out of reach before Apollonius* introduction of the two-hranchcd 
conception of the hyperbola. Securing the requisite lemmas is (he project in III, 
20-23. One would thus suppose that only the partial solution, perhaps along the 
lines of the method reconstructed by Zeuthen, was worked out by F.uclid and 
his followers, then later extended hy Apollonius. 

A puzzle attaches to the set of theorems which close Apollonius* Hook III. 
Like the propositions (16 23) just considered, the later set (Prop. 53-56) deal 
with relations of segments of secants and tangents drawn to conics. Consider 
111. 54 (see Fig. 14): if TQ. TQ # arc tangents to a conic. V the bisector of chord 
QQ\ and if to any point R on the curve lines are drawn from Q and O' to nice! 
in S’ and S, respectively, the line drawn from Q' parallel to TQ and the line 
from Q parallel to Ty\ then the segments ys, y’S' contain a given area; that 
is, ys * y'S' : yy" = <PV ? : PT ? ) cry * TQ' : QV‘). a given ratio.” 4 Zeuthen 
suggests that this theorem might play a role in Apollonius* own solution of the 






126 Ancient Tradition ul (iconic trie I'uiblcms 


t tic V iUiii.Mhui %«i i 



Figure 14 


three-line locus problem, and (his has been accepted by Heath and others.*' But 
this view can hardly be correct, since III, 54 is merely an elaboration of III, 16. 
while the latter itself suffices for a straightforward solution of this problem, as 
we have seen, Elsewhere. Zcullren notes lire patent projective character of III, 
54: for the constancy of the product QS • Q’S’ enables one to assign to each 
line in tire pencil at Q a unique line from the pencil at Q*. and this correspondence 
specifics a conic projcclivity in the manner of Steiner.*" To be sure, this element 
of the theorem is implicit in it. But even if one can now perceive here the “germ 
of the projective generation of the conic sections*’,*’ one can well doubt whether 
any ancient geometer would have conceived the same result in this sense.*" 
Indeed, Apollonius* theorem immediately suggests his intent to solve a specific 
problem: to determine a conic passing through three given points and being 
tangent at two of these to given lines. For the givens specify the value of 
QS • Q'S‘ : QQ'*. so that the ratio PV : PT is given from III. 54. ‘litis determines 
P and also the harmonic conjugate point P' (from PV : PT = P'V : P'T), so 
that the diameter PP’ is known. From the relation QV 1 : PV • VP' = L : PP', 
the latus rectum L is also known. We have thus found the required conic.** As 
before, the other propositions (55 and 56) permit solutions for other configu¬ 
rations of the problem, where both branches of a hyperbola are to be considered. 
In view of this straightforward interpretation of these theorems, Zeuthen’s pro¬ 
jective hypothesis Irecomes less persuasive. 



An interest in constructing conics in answer to such initial conditions of 
incidence and langcncy may be detected in other parts of the Comes. For instance, 
one learns in III. 41 that three lines tangent to a parabola cut each other in 
proportion: that is. PP' : P'T = P'R : RQ’ = TQ’ : Q'Q (see Fig. 15). The 
corresponding problem would be to determine a parabola tangent to three given 
lines and passing through a given point on one of them. Knowing any one of 
the points P. (J . R, the other two can be found via the relation in III. 41. If V 
bisects PQ, then TV will be the diameter of the parabola, and if S bisects TV. 
S will Ire its vertex (for PT is tangent, so VS « ST); thus PV’ - SV* I. de¬ 
termines its latus rectum L. In our discussion of Apollonius in Chapter 7, we 
shall find that lire detailed examination of problems of this sorl formed a major 
area of interest within his work."’ 

Euclid's efforts on problems like that of the three- and four-line locus thus 
initiated a fruitful activity of problem solving within the range of the developing 
theory of conics. The fragmentary character of the extant evidence limits one's 
ability to distinguish (Euclid's own findings and methods from those of the later 
geometers who continued his work. But our brief consideration here of samples 
from this field should make an important aspect of this work clear: that its 
concerns and methods were decidedly not projective. Zeuthen's perception of 
correspondences with the objectives of modem projective geometry seems rather 
to mistake technical coincidences for essential insights into motivation. Our hope 
to find within this material the context providing the rationale for the Euclidean 
Porisnu has thus been frustrated; for it is the markedly projective appearance 
of its subject mailer which most requires explication. The key we require will 
emerge, as we consider yet another of the lost Euclidean treatises, the Surface 
lutci. 



^Tradition ol Gcoiuutnc Problems 


A ANGLE TRISECTION VIA “SURFACE LOCUS'* 

iipus presents a series of three angle trisections, of which the third raises 
certain suggestions of an origin near the time of Euclid. We may sketch it as 
‘ follows (sec Fig. 16): vl 

Let it be required 10 cut the given arc AUG at H .such that arc HO is its third. 
(Analysis:) let it be done* so that angle BGZ is iwicc HA/,. Draw 01) to bisect 
angle BGZ, meeting BA at D. and draw DE and HZ perpendicular to AO. Thus, 

AO * DG and AE -EG. Since angle BGA is bisected, AG : BG = AD : DB 
(Elements VI. 3); but AD;I)B = AE : EZ (via similar triangles), so that 
AG: AE = BG : EZ = 2 : I. Thus, B lies on the locus such that BG - 2EZ, 
for given points G, E; that is, the Iikus which is such that HZ 1 *♦ ZG i : EZ' - 4 
: I, This is a hyperbola, given in position, so that its intersection with the given 
arc ABG solves the problem. And the synthesis is obvious. 



One secs that the angle trisection has been reduced to the specification ot a 
hyperbola via focus and directrix, that is, as the locus of points whose distances 
from a given point and a given line arc in a given ratio greater than 1:1. But 
we note (hat the locus is first transformed into one involving ihe squares ol those 
distances, rather than the distances themselves. Elsewhere, Pappus presents a 
proof of this locus construction, and there loo the problem is expressed in terms 
of the squares of the distances.'*' It is part of a general account of the focus- 
directrix specification of all three conics which Pappus provides as lemmas to 
Euclid's tjH'i an Surfaces"' Since the angle Bisection is incomplete without this 
lemma, and since it conforms with the special formulation of the problem ap¬ 
pearing in it, we must suppose that both bore some relation to this lost Euclidean 
analytic work. 

We face the problem, however, that nearly a century after Euclid, the parabolic 
case of this locus is constructed by Diodes without indication of awareness of 
a prior treatment.*” Although Diodes might merely have been ignorant of Eu¬ 
clid’s treatment, it seems better to suppose that he did indeed play a part in the 
discovery of this locus. But what part? Certainly, if the treatment of the parabolic 
case given by Pappus were known earlier, the solution of Diodes, which is 
essentially identical to it, could hardly have been original. Is it possible, then. 


that an alternative method was modified by Diodes, thence to serve as a basis 
for the version given by Pappus? 

A consideration of the Euclidean work will assist us. What was a “surface 
locus’*? It is natural to suppose that it had to do with the tracing of curves on 
given surfaces, as via the intersection of two surfaces, and this view is in accord 
with rcniatks by Pappus and Piuchis.”' If so. we may perceive in the cube 
duplication of Archytas and the l»ip|Htpcdcs of Eudoxus instances of such curves, 
and the former enters as a locus directed toward the solution of a problem. Other 
instances arise in Pappus: one may trace spirals on the surface of cylinders, 
cones, and spheres via projections of the plane spiral. While one usually assigns 
the deliiiit ion of the latter to Archimedes, it is clear from Pappus that this curve 
was known to Conon somewhat earlier, so that its introduction and first study 
might well have occurred in Euclid’s time/"* To illustrate surface locus. Pappus 
describes the curve formed by intersecting a plecvoid surface (i.c., a spiralling 
ramp) by an oblique plane: the orthogonal projection of the plane curve so lormcd 
is the quadralrix/” Pappus* method is analytic: assuming E on the quadratrix 
AK, the ratio EZ : ate DCi (= AB : arc AG) is given; for tl on the spiral, the 
ratio OD : arc !)(!(= AM : arc AG) is given (sec Eig. 17). Setting El = OD, 
the ratio of lit : EZ is thus given (Pappus does not state its actual value. AM : 
AB), so that I is given as on the intersection of the surface with a given plane 
through Zl. One may note that this alternative form, like the “mechanical” 
mode it would supplant, still requires appeal to continuity for determining its 
terminal position, the key to its use for the circle quadrature/”* Doubtless, the 
projective form arose in connection with other applications of the curve, such 
as its use for the division of angles. Euclid himself is likely to have known of 
such uses through the efforts of predecessors like Diuostratus. 







130 Ancient Tradition of Geometric Problems 




<h) 

Figure 18 


A 



Using a mctluHl directly suggested by the solid projections presented by 
Pappus for the quadratrix, we can construct a solution to the focus-dircctrix 
problem in the following manner: first consider the case where the given ratio 
is I: I, the given points A and B; we wish to find the locus of P such that 
AP = BQ, for Q. the fool of the perpendicular to AB from P (see Fig. 18). 
Let the square ABCD lie in a plane perpendicular to the given plane, and the 
line EF in a plane parallel to the given. Then the locus of points at distance AF 
from F is a circle which meets AC, the diagonal of the square at (•; while the 
locus of points at distance EB ( = AF) from the line through E perpendicular to 
BC in the elevated plane is a line parallel to that line meeting the diagonal HI) 
at H. If we now consider ail positions of the horizontal EF, the first locus will 
be a cone with vertex ai A, axis AD, one generator coinciding with AC. and 
the vertex angle a right angle; while the second locus will be a plane passing 
through the line BD and containing the line perpendicular to it in each elevated 
plane. Since the lines BD, AC meet at right angles at K, the intersection of the 
surfaces will be a parabola, “the perpendicular section of the right-angled cone,” 
and if we express it as Y* = N-X via coordinates in the oblique plane, then the 
line of application N will equal 2DK, as we saw above. Now, if we project this 
parabola orthogonally onto the original plane, lines AP and BQ will be equal, 
for the associated lines FP' and EQ' (that is, EH), respectively equal to them 
arc equal to each other in the elevated plane. Thus, P lies on the required locus. 99 
Moreover, the projection leaves Y unaltered, but shrinks X by the factor 
I : V2, so that the curve will be expressed by Y' 2 = N' • X\ for N' = 
2\/2 AK or 2AB. Thus, the required locus is a parabola of parameter twice the 
given line. 


The eases of the hyperbola and the ellipse are analogous, each utilizing a 
rectangle ABCD in place of the square used for the parabola. Let AL : LB be 
in the given ratio, and draw the right triangle ABK such that KL is the altitude 
to the hypotenuse AB (see Fig. 19). Then if we complete the rectangle ABCD 



Figure 19 







such that AKC is ils diagonal, <AB : BC)* = AL ; LB, the given ratio. We 
designate the line BB' perpendicular to AB in the base plane perpendicular to 
that of ABCD. If we now introduce the cone of axis AD and generator AC, it 
will intersect the plane of the lines BK, BB' in a conic section, each of whose 
points is such that its distances from the line AD and the plane of BC\ BB' are 
in the given ratio AL : LB. Setting K' as the point where BK extended again 
intercepts the cone (in the ease of the hyperbola, its opposed branch) and hy M 
the point where BK meets the axis Al), we have the diameter KK' and the 
parameter 2KM for this conic, such that the coordinates X, Y for each of ils 
points satisfy the relation Y* : X (KK‘ X) = 2KM : KK'. If we project this 
conic onto the base plane, the projected ordinate Y' = Y, while the projected 
abscissa 


Mi 

X' = (—) X. 

UK 

The points on this projected curve satisfy the locus condition that their distances 
from the point A and the line BB' arc in the given ratio AL ; LB. Since 

Y’ } : X' (LL* S. X’) (—V = 2KM : KK’ = 2AI. : LL\ 

HI. 

and KB*’: 51/ = AB ; BL, it follows that 


Y : X' (LI/ .• X ) = 2AB ( -) : LI/. 

L« 

Thus, the curve solving the locus is the conic of diameter LI/ and parameter 

2AB (—1. 
i.H 

The view lhal Euclid’s Surface Loci took up the solution of problems via the 
intersections of solids, in u manner like that given here, raises interesting pos¬ 
sibilities concerning the nature of the I*ah sms. Consider, for instance, that in 
the above construction each of the lines HI.', BK' is divided harmonically; that 
is. BL' : BL = Al.' : Al. and BK’ : BK = MK’ : MK. Several of Pappus’ 
lemmas to the Porisms intend to show that this harmonic relation is preserved 
through projection (Lemma 19; cf. 5. 6). Thus, one can see how such lemmas 
on solid projections might find their use within the study of geometric problems. 
The harmonic relation is especially important for the investigation of the tangents 
to the conics. Indeed, hy virtue of this division in the ease of lines BL'. BK' 
above, it follows that the tangents drawn to the conics al the |H>ints corresponding 
to the abscissas AL. KM will intercept their respective diameters at the point 
B. This properly of the tangent lines is proved by Apollonius in Conics I. 34 
(cf. 33 for the parabola). Imt one can conceive an alternative derivation via solid 
methods. First consider the ease of a circle lo which arc drawn Ihe two tangents 
KA, KA’ |scc Fig. 20(a)|: if any secant line from K meets the circle in P. Q 
and the chord AA’ in N. then KP : KQ = NP : NQ (or equivalently. KN is the 


Figure 20 


harmonic mean of PK. Kyi."” Pappus includes a proof of this among (he lemmas 
to the Porisms (Lemma 28).“" Let us next consider the case where PQ is a 
diameter of the circle, and let us view the circle as the section of a right cone 
of vertex B |sec Fig. 20(h)). Then if the figure is projected onto an oblique plane 
from the center B lo yield a conic of diameter RS. the projected lines 01.. G'L. 
corresponding to KA. KA'. respectively, will be tangents lo this curve (for 
incidence relations are preserved), and the line I.RMS will be divided harmon¬ 
ically in correspondence with KPNQ." ,J Hence, the harmonic property of the 
tangents to conics follows as a projective consequence of the property for the 
circle. At the same lime, the harmonic division of any secant to a conic likewise 
follows from the ease of the circle. 

With reference to the same circle, we may ask for the division of secants 
corres|x*nding to the geometric and arithmetic means (see l ; ig. 2l(a)|. Since AK 
is the geometric mean of the segments PK, KQ of any secant, it lollows that 
the locus of points J on the secants such that KJ is the geometric mean of PK, 
KQ will be Hie circle or center K and radius KA. The points I which mark off 
Kl as arithmetic mean of PK, KQ will be the midpoints of Ihe chords PQ, and 
hence lie on the circle of diameter OK (for 0 the center of the given circle), since 
the angle OIK will always he a right angle. If now, as before, the circles are 
conceived as sections of a solid, they can be projected onto oblique planes and 
so yield determinations of the conics whose points divide the secants to given 
conics according to the specified conditions of harmonic, geometric, and arilh- 




Figure 21 


metic division |scc Fig. 26|. ,<M Later wc shall encounter a speculation by Zeuthcn 
assigning to Bratosihcnes an inquiry into locus problems of this type. The present 
remarks show how naturally the investigation of solid projections could develop 
in this direction. 

Of these three locus propositions with reference to the circle. Pappus gives 
only that one relating to the harmonic property, but not those relating to the 
geometric and arithmetic means. Nevertheless, he does provide, in two other 
lemmas, certain other probities of a comparable kind (see Hg. 22): 

Lemma 33: If points N, K lying in the line of the diameter of a circle arc such 
that NK is lltc geometric mean of PK. KQ, ami K' is any point on the line 



Figure 22 


perpendicular to QK ai K. then any secant K'P'Q' passing through N is divided 
such that NK' is the geometric mean of P'K\ K'Q’. 

Lemma 35: If in the same figure it is assumed that NK is the harmonic mean of 
PK, KQ. then also NK' will be the harmonic mean of P*K\ K'Q\ ,,M 

One notes that just as the earlier lemma (28) on the harmonic mean produced 
the construction of the polar line with respect to the pole K lying outside the 
circle, so in l^mma 35 one has the construction when the pole N lies inside the 
circle. In both eases one can obtain via solid projection in the cone an alternative 
proof of Apollonius* corresponding propositions for conics (111, 37-40). 

Again consider the section of a right cone by a plane perpendicular to a 
generator (see Fig. 23). We have seen that if the diameter of the section PQ 
meets the axis of the cone in N and the base plane in K, then the line KPNQ 
is divided harmonically; and that if AN is an ordinate, the tangent at A passes 
through K. Now let any other plane containing line KK' perpendicular to KB 
in the base plane intersect the cone in the section of diameter P'Q\ Using B as 
the center of projection, wc find that KP'N'Q' is also divided harmonically. 
Thus, if A' is the projection of A, the line KA' is tangent to the projected conic. 
The converse suggests a problem: to find the locus of points of tangency A 



Figure 23 






losing mruugii a £•**.«» iiiic rwiv . v/iic a|vpio<tt;ii iiugni uihmuci me prujauun 

of one section P0 onto another P'Q' via the center 13. As we have seen, since 
KPNQ is divided harmonically, so also is KP'N'Q'. Since KA, KA' are tangents, 
Ihe lines AN, A # N' are each parallel to KK' and hence to each other. Since, 
furthermore, AN, A'N' are radii of the circular sections of the cone made by 
planes parallel to the base through points N, N\ respectively, it follows that 
AN : A'N' = BN : BN'. Hence, the triangles BAN, BA'N' arc similar, so that 
BAA' is straight. Thus, the sought locus of points of tangcncy A is the generator 
of the cone BAA' extended. One may note the corollary, that if the lines AP, 
A'P' are drawn and extended, they will meet at a point on KK'; for the triangles 
APN. A'P'N' arc in projective correspondence in accordance with the config¬ 
uration of Desargucs. 1 "' Thus, for all curves in this family of sections, the line 
AP will pass through the same point K" on line KK’. 

These observations on the possible nature of Euclid's investigation of surface 
loci suggest a route toward the resolution of several puzzles surrounding his 
Porisms. In view of Pappus’ lemmas to this work, wc would gather that Euclid 
cither assumed them without proof or else used a different method of proof from 
that given by Pappus. The context of stereomctricnl problems offers an appro¬ 
priate set of techniques for such alternative proofs in contrast with the metrical 
procedures of plane geometry adopted by Pappus. Furthermore, the lemmas 
reveal the affinity of the subject matter of the Porisms with that commonly 
encountered now in the field of projective geometry. Yet it is clear that the 
ancients never introduced the conceptions and techniques of projectivity fun¬ 
damental to the modem field, and so could never have undertaken the systematic 
exploration of the relations organized within that field. What, then, were the 
initiating interests which led them into these materials? Wc have seen how the 
study of problems relating to conics, as for instance the drawing of tangents, 
could give rise to configurations significant within the projective field. Among 
Ihese arc the two-triangle ligurc of Desargucs, the complete quadrilateral, the 
harmonic division of a line and the preservation of this division under projection. 
For these have natural interpretations in the context of problems involving the 
formation of conic curves as different planar sections of the same cone. 

If through his Porisms and Surface Loci* Euclid did indeed attempt to in¬ 
augurate a stereometrical study of the conics, his approach was soon modified 
drastically. This three-dimensional aspect is of course still in evidence in Ar¬ 
chimedes’ studies of the conics of revolution; but they have all but disappeared 
from Apollonius' theory of conics. For here the formation of these curves as 
conic sections serves only to fix their conception and to permit the derivation 
°f ihc planimetric second-order relations of their ordinates. Thereafter, all prop¬ 
erties and applications of the curves follow from these relations via the manip¬ 
ulation of proportions of lines in the plane. 1 "'’ For instance, the problem of the 
locus relative to a point and a line, which wc solved above through the sections 
of a cone, is solved planimetrically in an alternative method preserved by Pappus; 
here the condition of the locus is transformed so that either Y* = 2AB • X or 


n U MUM IIIV £IV«rll I Ml IV flU • •• • UMHVWH »V J/v I w I . V iivfr u mvi wimv,mv 

treatment comparable to that reconstructed above might have served as a prec¬ 
edent for the extant planimetric treatment. If Euclid had solved the locus problem 
in the former manner, perhaps even omitting the parametric expressions for the 
I solving conics, there would have been ample cause for Diodes and for the author 

of the lemmas to the Surface Loci to produce their alternative solutions. In this 
way, wc can reconcile Euclid’s awareness of the focus-directrix property of the 
conics with their re-examination by these later writers. 

EUCLID’S CONTRIBUTIONS TO THE STUDY OF 
f PROBLEMS 

None of the lost mathematical writings alluded to in ancient sources seems more 
enigmatic than Euclid’s For isms. Pappus* extensive lemmas and commentary 
on the work have permitted detailed and persuasive reconstructions, especially 
that by Chaslcs. But its underlying motive and the scope of the wider geometric 
field toward which it was compiled remain unclear. The ambitious view advo- 
( cated by Zculhcn seizes upon the striking correspondences with the modem field 

of projective geometry to assign to Euclid and his followers the elaboration of 
a projective theory of the conics, culminating in the projective generation of 
conics with reference to live given points, a result equivalent to Pascal’s “mystic 
hexagon.” Unfortunately, the ancients never introduce the general conceptions 
essential for the rationale of projective geometry. I have attempted an inter¬ 
mediate position, suggesting that Euclid experimented with a stereometric ap* 
t proach to the study of conics. His own Optics provides a ready context for such 

an inquiry; for instance, his observation that obliquely viewed circles appear 
“pressed in” (Prop. .16) at once raises the question of the precise determination 
of the apparent curve, whose solution as an ellipse actually appears among 
Pappus’ lemmas to this work. 10 ' My view would thus assign to Euclid a form 
of the study of conics not unlike that later adopted by Desargucs and Pascal, in 
which pro|Hisitions established for circles arc extended to conics via solid pro¬ 
jection. um llris dt>cs not intend to deny that the projective conceptions were for 
the most putt absent from the ancient work and that these were thus truly novel 
elements of the 17th-century efforts. Nor do the several reconstructions of conic 
theorems given above presume to exhibit the method actually adopted by Euclid 
and his followers, hut rather serve as specimens of a field of results readily 
accessible to them upon the adoption of such a stereometrical approach. The 
fact that Desargucs and Pascal, firmly grounded in the study of Apollonius and 
Pappus, moved in tin’s same direction suggests that they might in part have been 
rediscovering forgotten aspects of the ancient work in the process of advancing 
their own. Our task of interpreting the ancient field will surely have a more 
appropriate instrument in these 17th-century efforts than in the more sophisticated 
studies by the 19th-century projective geometers. 

It is thus clear that Euclid’s study of the conics, his Parisms and Surface 




/-or/, together with his Data and Element*, had special utility for the analytic 
investigation of geometric problems. The study of locus problems appears to 
' have attracted special interest at this time. We have seen examples in the two- 

> point locus of the Aristotelian Meteorologica , the point-line locus (or the spcc- 

\ ification of a conic via focus and directrix, as in the lemmas to the Surface Uni). 
£ and the locus with reference to three and four lines; doubtless many more of 

1 comparable type were then examined, although our sources no longer preserve 

t clear information on the wider scope of this activity. 

£ As for Euclid himself, one surely docs better to view him as an effective 

t. teacher and compiler than as a phenomenally gifted mathematical intellect in his 

J own right. The Elements . for instance, drew on the discoveries by Theaetetus 

'l and Eudoxus and their followers, while its individual books appear to have 
' already received a systematic organization in the decades before Euclid. The 
; was novel in form only, its content largely duplicating that of the Elements , 

; but cast specifically for application in analyses. In the area of the conics, Euclid 

r had the work done by Aristaeus as a precedent. We must regret not knowing 

more about the Surface Loci and its precursors; but. presumably, work extending 
as far back as Archytas and Eudoxus, in their studies of the cube duplication 
and other solid constructions, could serve as background. As for the Porisms , 
however rich the potential field might have been relating to it, Euclid’s treatment 
had its weaknesses. For instance. Pappus was able to formulate into single 
propositions what Euclid presented in as many as ten and could sec how some 
of Euclid's results extend into more complicated configurations in a way not 
noted by Euclid. MW The proliferation of special cases which seems lo have marked 
Euclid's approach must surely have obscured the general patterns in this material. 
But Apollonius charges Euclid not only for having taken up merely “the chance 
portion" of the locus of three and four lines, but also for having done so 
"unsuccessfully." These criticisms are implicitly admitted by Euclid's apolo¬ 
gists, in a passage cited by Pappus, in that they insist lie made no presumption 
of having exhausted these fields, but that his effort proved to be an indispensable 
basis for further research, including that of Apollonius himself. mi 

Wc should keep in tnind, then, that in Euclid’s time the theory and application 
of the conics were still relatively new, constituting a field undergoing its first 
consolidation. His contributions were seminal for the development of problem 
solving in the 3rd century. But it is entirely unfair of us to demand that his own 
efforts here matched the refinement of the Elements or of the Apollonian Conics. 

NOTES TO CHAPTER 4 

' The biographical data on Euclid arc meager. Ilis activity at Alexandria is attested 
in passages by Pmclus and Pappus; but the former is anecdotal (“no royal road to 
geometry" addressed to King Ptolemy), while the latter refers lo Euclid's students at 
Alexandria (cf. T. L Heath, Euclid's Elements , I, Ch. I for references and discussion). 
On the other hand, the nature of Euclid’s work well suits the environment of Alexandria: 
his several textbooks on geometry and mathematical science would firul a welcome place 
in the research and teaching ni the Museum and use to the fullest advantage the textual 


resources of the Library. One usually sets his activity near the time of the founding of 
the Museum, thus centering on the first two decades of the 3rd century n c Note, however, 
that Archimedes* earliest recognizable citation of Euclid's Elements is in Spiral Lines. 
well after the middle of the century, so that a later dating may be indicated for Euclid 
(see my “Archimedes and the Elements" and “Archimedes and the pre-Euclidean Pro¬ 
portion Theory"). Recent authors wish to deemphasize the role of Alexandria as a center 
for mathematical anti astronomical study; cf. (i. J. Too me r. Diodes, p. 2 and O. Ncu- 
gchaticr. History of Ancient Mathematical Astronomy, pp. 571 f. 

• Heath. History of Greek Mathematics, I. pp. 357 f. Sec also his survey of major 
editions of the Elements from antiquity to the early 20th century in Euclid's Elements. 
I. Ch. VIII. In addition lo Heath*s ample notes on the Elements, one may consult the 
studies by E. Ncucnschwander. "Die ersten vicr Bucher dcr Elemente Euklids." Archive 
for History of Exact Sciences, 1973. 9. pp. 325-380 and “Die stercomctrischcn Bucher 
dcr Elemente Euklids." ibid.. 1974-75. pp. 91-125: and I. Mueller. Philosophy of 
Mathematics and Deductive Structure in Euclid's Elements. Cambridge. Mass.. 1981. 
The background to the Euclidean theory of incommensurablcs is examines! in my £ty#/irfrwi 
of the Euclidean Elements. Dordrecht. 1975. while a survey of Rook X of the Elements 
is provided in my “I.a Croix dcs Mathematicians.“ 

’ On this technique, see my Evolution of the Euclidean Elements. Ch. VI. pt. IV. 
The standard account derives from H. (». Zculhcn; sec also Heath. History. I. pp. ISO- 
154 and Apollon at s of Perga, pp. cii-xi. 

' An account of the relation of the Elements to prior efforts appears in my Evolution . 
Ch. IX. with references there to other discussions. 

Proclus. hi End idem, cd. Erictllcin, p. 426. Sec Heath s account of the theorem 
in Euclid. I. pp. 350-354. On its Babylonian appearances, in particular in the “Pytha¬ 
gorean triplets" on die cuneiform tablet Plimpton 322. see O. Ncugcbaucr. Exact Sciences 
in Antiquity, pp. 35-40 and B. L. van dcr Wuerdcn. Science Awakening, pp. 76-80. 

See my "Archimedes and the Prc-Euclidcan Proportion Theory" and Exhalation. 
Ch. VIII. Pt. II and Appendix B. 

On irrationals see my Evolution. Ch. VIII. Pt. IV. On the pre-Hue I id can study of 
the postulate, see I. Toth, "Das Parallelenprohlcm ini Corpus Arislotclicum." Archive 
for History of Exact Sciences . 3, 1967, pp. 249-422. 

Aristotle, Opera 4lickkcr). 376 a 4-6. I have altered the punctuation slightly. Note 
that the writer tacitly assumes that MK “ Kll dicing radii of the same circle) in order 
lo infer that the ratio 1IM : MK is given. 

Apollonius* proof, to be discussed below, is given in full by Eutocius in his 
commentary on the t'onux: sec Apollonius. Opera . cd. Heiberg. II. pp. 180-184. That 
this result was proved by Apollonius in the second book ol the Plane bn i is asserted by 
Pappus. Collection (Hook VII). cd. Hultsch. II. p. 666. Two detailed accounts, estab¬ 
lishing the identity of the two methods and giving full technical and textual comparisons, 
arc presented by Heath, the first in his Euclid, II. pp. 197-200 (commenting on VI, 3). 
the second in his Mathematics in Aristotle (1949), pp. 181-190. Ihc latter accounl was 
motivated by Heath's regret that writers on die Aristotelian meteorology had overlooked 
his earlier analysis and so ened in their discussion of this passage. It is to be all die more 
regretted dial the Loch translator of the Meteorologica repeated the oversight a few years 
later (1952). I\ Solmscn’s hesitation to accept the accuracy of Heath’s position on this 
passage is likewise rcgreUnhtc {Aristotle's System of the Physical World, I960, p. 420n). 


*“ I follow (lie construction in the Meteoroloyica, but have partially adapted the latter 

of the proof where it seems muddled and have avoided its use of an indirect mode 
of reasoning. 

” It is possible that this slip, together with the adoption of the indirect mode of 
proof, might indicate the author's having confused this construction with the proof of its 
converse, the completeness of the solution. For the proof that all points satisfying the 
condition of the locus lie on this circle is indeed well soiled to an indirect demonstration. 
as one secs from the treatment given by Apollonius. Note further that the point n here 
cannot be the same as N in Fig. 4. even though the author seems to view it so. Apparently, 
he has confused two senses of the phrase "the arc through M." 

11 Heath notes this detour ( Elements , II. p. 200) without registering surprise at the 
monumental lapse implied in (he Apollonian treatment. 

Heath summarizes Sinison's method in both of his accounts. 

M This relation, not in the Elements , is assumed by Pappus in the CoUecthm (Hook 
VII). II. p. 730. 

Oddly, Heath persists in following Sinison's approach in attempting his own re¬ 
construction of the analysis underlying the Meteorolo^aa passage. Furthermore, he seems 
to assume that one knows beforehand that the locus will be a circle. Hut surely, this fact 
too must be revealed by the analysis, not just its center and radius. These lapses detract 
from (he effectiveness of his account for displaying the line of thought actually underlying 
the Arisintclian/Apollonian approach. 

It is remarkable that Pappus adopts much the same method in the solution of a 
related problem. In Lemma 29 to F.uclid's Porisms {Collection (VII). II, pp. 904-906). 
he seeks to inscribe a broken line ACiH within a given segment of a circle, such that AG 
• GB is a given ratio (Fig. 24). Thus, lie draws the tangent line GD, whose length is the 
mean proportional between BIX DA. whence AD : DB = AG' : GB*. Since AG GB 
is given, so also is AD : DB; since I) is thus given, so also is G. Here, the tangcni hud 
corresponds to PM in the Aristotclian/Apollonian locus problem. In effect. Pappus’ 
method finds the solution via the intersection of two loci: that answering to the given 
ratio and that answering to the given angle of separation of AG, GB (i.c.. the given 
circular arc). As il happens. al-Biruni solves Ibis same problem in his Hook of Chords 



Iff. IL Stilcr. "Illicit dcr Aul I indung der Sehnen i in Kreise von.. .el-Biruni," Hibliotheca 
MaihenuOica, 1910-1911. 11, p. 36); since the angle is given, he shows one need only 
construct a triangle with this angle nt its vertex, such that the adjacent sides arc in the 
given ratio, and then inscribe (he similar triangle in the given segment. That Pappus 
follows such an elaborate alternative method here must indicate that he has lifted this 
solution from a context which treated it as a corollary to the two-point locus problem, 
the latter presented in the Apollonian manner. This may indicate that Euclid himself in 
(he Porisms presonlcd the locus problem in this same manner. 

” The proof that this locus is a circle appears earlier in the account of the rainbow. 
Meteor. Ill, Ch. 3. 

*" Alexander of Aphrndisias (2ml century a i> ). In Meteorofoxica. cd. Hay duck 
(Commentanain AristotelemGraeco. Ill), pp. 162 -172;Olympiodonis (6th century a r>), 
In Meteor , cd Sliivc (CVUL XII). pp. 243-259. Note that both commentators assume 
the simpler version of the uniqueness proof just given as Aristotle's intent (Olympiodorus, 
txsson 38. pp. 245 If). Hiey should thus view the entire remaining portion of the geometric 
proof as superfluous. Yet each goes on to an elaborate line-by-line exegesis of that proof, 
without seeming to grasp what that proof is intended to establish (ibid., Lesson 39, pp. 
253- 258). Note that some modem translators and commentators appear to be in the same 
situation (see note 9 above). 

lliis aspect of the passage was brought to my attention by H. Mcndell. 

Pace Soli men, who appears content with assigning the proof to Aristotle himself 
{op. p 420). 

I touch on these passages in "Archimedes and the Pre-Euclidean Proportion The¬ 
ory” and m an unpublished essay. "Zeno, Aristotle and Eudoxus." But they have not 
received serious attention among Aristotle scholars, including Ross {Aristotle's Physics) 
and Hcalh {Mathematics in Aristotle) Cf. my review of II. J. Wasehkicss, Von Kudos os 
r« An sauries, in Isis . 1980. 71. p. 507. 

'* Cl. the pseudo-Euclidean Caioptrtca. Prop. 30 (Euclid. Opera , cd. Heiherg, VII) 
and Diodes' Hunting Mirrors, cd Tonmer. Prop 1-3; here one investigates burning 
rniirots by which rays fiom the sun are reflected to a point oi a line. 

*' Cf. Solmsen. op. cit ., p. 419, 

M C. Boyci, I he Hat a bow: burnt Myth to Mathematics . 1959, pp. 39-54. 

*' That is, when the sun is rising or setting at the time the rainbow is seen. Generally, 
one must lake lay Mil parallel to the direction of the line from the observer at K to the 
sun. 

* See H I). P. l ee's introduction to the Loch edition, pp. xxiii- xxv. Note that the 
authenticity of Hook IV. on the nature of chemical processes. has long been doubted, 
although \jcc defends il {ibid., pp. xiii-xxi). 

Kuclid also standardizes "right line" (eutheia) in preference to "line" (grrimmi 1 ) 
for rectilinear segments, and is followed in this by later authors. 

M On Hippocrates, see Chap. 2 almvc. Resides the Prior Analytics , one may consult 
Aristotle's Physics (IV, ch. K; VI, ch. 2, 4-7. 9) and de Caeto (1, 12; II, 4; 111, 2) for 
this usage. Arming later authors, only Philo of Byzantium adopts the same, to my 
knowledge. A mechanical writer late in the 3rd century nr, his account ol the cube- 
duplication bears several indications of his distance from tlie field of geometric research 
at that lime; the passage in Marsden (Greek Artillery, p. 110) may be compared with 





h accounts by Hero, a mechanical writer, like Philo, but much closer to the standard forms 

f of geometrical writing fcf. Pappus. Collection , I. pp. 62-64). 

n Sec note 11 above. 

* Although we of course cannot know for certain the identity of the geometer who 
discovered this solution of the locus problem, it may well be pertinent to (his question 
that Proclus says of Hcrmoiimus of Colophon, a bit before the time of Euclid. that he 
“composed some things on loci** (//i EucE, p. 67). 

" For references to Pappus and Proclus on the meaning of “purism.” sec Heath, 
History of Greek Mathematics, 1. pp, 431 f, 434 f. 

11 Pappus* commentary on this corpus occupies the entirety of the 238 propositions 
of the seventh book of his Collection . and is our principal, often unique source as to its 
contents. 

11 The Data has been edited in Greek by Mcnge (Fuelid. Opera, VI), and its medieval 
Latin translation has reccnily been edited with English translation by S. Ito. 

u Sec, for instance. Heath's discussion of Prop. 93 [His ft try, I, pp. 424 0- 

11 For authorities espousing this view, see Ito, op . Wr. pp. 11 I . But most com* 

mentators appear to have had a sense of the relevance of the Data for analyses; cf. Heath, 
History , !, p. 422. 

* Aristotle notes that propositions arc not always simply convertible, and secs that 
as a cause why analytic methods arc more useful in geometry (whose propositions tend 

) regularly to be convertible) than in dialectic (East. An . I, 12). For a logical and historical 

/ account of this method, see J. Hintikka and U. Kernes, The Method of Analysis, 1974 

and M. Mahoney, “Another Look at Greek Geometrical Analysis.” Archive for History 
l of Exact Sciences, 1968/69, 5, pp. 319-348. See also Chapter 3. 

s * 

f ” Proclus informs that Ixon, a contemporary of Plato, investigated diorisim to 

£ determine the limits of the solvability of problems (In Enel idem. p. 66). This bears on 

I the Meno passage discussed in Chapter 3. 

\ ** See, for instance, the striking similarity between Data, Prop. 58 and Elements VI, 

* 26 on the application of areas. On the other hand, some parts or the Data adopt a mode 

at variance with the Elements. The section cm parallels, for instance, conceives of a 
“displacement** intetapiptein) of given lines (Data, Prop, 28) in contrast with the strictly 
static view in the Elements (I, 30). It may thus he that some portions of the Data hud 
already been compiled before Fuel id and were incorporated by him into a larger treatment, 
with his own additions based on the Elements. 

" Cf. the problems in Elements I. I -3. 9-12, 22-23, 31, 44 46; all of IV; further 
instances in VI, XI. XIII A comparable usage occurs in the Aristotelian tract On Indi¬ 
visible Unes (970 a 8): to construct a triangle whose three sides arc “given** (cf. Elements 
I, 22), 

4,1 De Caelo 1,2, 283 a 6; for other instances, sec Bon it/.' index in Aristotle, Opera 
(Berlin edition). V, s.v. didonii. 

41 Collection (Book VII). II. p. 672; cf. p. 676 and below, note 57. We return to 
Aristacus in Chapter 7. 

41 Cf. Quadrature of rite Parabola, Prop. 1-3. A survey of such instances is given 
by Heath. Apollonius, Cli. III. An effort to compile the whole of the Archimedean theory 
of conics was made by J. L. Heiberg. “Die Kenntnisse dcs Archimedes fiber die Kegel* 


schnitte.** Zeitschnft fur Mathematik und Physik (Itist.*litt. Abth.). 1880, pp. 41-67. But 
its comprehensiveness has recently been set into question (G. J. Toomcr, Diodes, p. 4). 

41 Sec Apollonius. Conics I, preface and Heath's discussion, Apollonius . Cli. 1. 

** Note that some parts of the earlier theory might happen not to appear in Apollonius. 
Such seems to he the ease for the theorems on the focus-directrix construction of (he 
conics (see Seel, iv below); cf. also Dioclcs* assumption of the constancy of the subnormal 
of the parabola, only found in Apollonius' Book V in somewhat different form (cf. 
loonier, op. rir., pp. 17, 151). 

For instances of such inquiries, see Chapter 3. 

** “On the Astronomical Origin of the Theory of Conic Sections,*‘ Proceedings of 
the American Phdosophtcal Society, 1948. 92, pp. 136-138. 

,r Collection (Book VI). II, pp. 588 ff; see Heath. History , II. pp. 397 ff. 

4 * | Aristotle |. Problems X V. 7. X VI, 6; discussed by Heath. Mathematics in Aristotle, 
pp. 264-267 Note that thyreos was si ill admitted as a name for the ellipse by Geminus 
(1st century ad ), as cited by Proclus (In Euclidem, pp. III. 126). In his work on 
spherical astronomy, the Phaenotnena, Euclid argues the sphericity of the realm of the 
stars on the basis that it always appears to produce a circle when sectioned by a plane, 
while in the case of cylindrical ami conical figures the result would be an ellipse, (hat 
is, “the section of an acute-angled cone** (Preface, Opera, VIII, pp. 4-6). Euclid is 
surely mistaken in alleging that (he appearances of these three sections will be different; 
for Ihc observer will view them from within the plane of their formation, so that the 
sections of the cylinder and the cone will appear precisely superimposed over the section 
of the sphere, giving all three the appearance of circles. It is surely significant that Ptolemy 
docs not use this argument in his account of the sphericity of the cosmos (,Vvtrfri.w.v I. 3). 

” Cf. Pioclus. In Euclidem, p. 356: one deduces the “symptom** (property) of the 
curve as the basis of its study. This characterizes Archimedes' treatment of spirals (Spiral 
Unes . Prop. 2, 12; Pappus, Collection (IV). I, p. 234), as also Nicomcdes* of the 
conchoids (Pappus, ibid., p. 244) and that of the quadratrix (ibid., p. 252). This distinction 
between generating mode and property has been exploited by Zcutheii in his argument 
(hat ihc ancients intended problems of construction to serve as existence proofs f“l)ic 
geometrischc Construction als *HxiMcn/bcwcis* in dcr antiken Geometric,** Mathenuh 
tische Antuden, 1896, 47, pp. 222-228); I raise reservations concerning this view in 
Chapter 8, 

Such a construction was used by J. Werner in his Elements of Conics (1522). 
Prop. XI; sec M. Oagcll. Archimedes in the Middle Ages, IV, Pi. |, pp. 254, 277. 298. 

41 Dioclcs and Fuiocius describe such pointwisc proccdmcs; see Cli. 3. notes 36-37. 

v Pappus. Collection (VII). II. pp. 672 -674; Fuiocius. In Apollonium, cd. Heiberg. 
II. pp. 168-170. Note that even those wriicrs like Kuclid and Archimedes who subscribed 
to the older mode were still quite aware of the nahne of oblique sections; cf. Heath, 
Apollonius, pp. xxxvi. xlv, and also note 48 above. 

See Toomcr. Diodes, pp. 9, 14. Geminus. and after him Puk'Ius. sometimes use 
the older terminology, perhaps a suggestion of their reliance on early sources; cf. In 
Euclidem, p. Ill 

M For a similar intuition of the role of Ihc solid construction within Ihc early theory 
of conics, see van dcr Waerden, Science Awakening, p. 245 and II. (j. Zenthen. Lehre 
von den Kegehchnittcn, pp. 467-469. Hie view is also sketched in Zcuthcn*s “Con* 



Mruction uls Fxisten/bcu’eis" (sec nine 49 above). One should no(c that all these allegedly 
aim lo provide a rationale for the Menaechmean conception of the conics. 

” This interpretation of the expression conforms with the view of Zeuthen and Meath; 
cf. Heath, Archimedes, pp. clxvii-iii; and Toomcr. op. cit.. p. 13. 

* Cf. Toomcr. op. cit ., pp. 5-7 and Heath. Apollonius. Ch. Ill and pp. Ixxix ff. 

” Pappus, Collection (VII). II. p. 674; cf. Heath, Apollonius, pp. Ixxxiii- iv. 

One may consult the derivations given by Toomcr {op. cit.. pp. 10—13) and H. J. 
Dijkstcrhuis [Archimedes . pp. 58. 59) for a proof that line C, as shown in Figs. 6(b). 6 
(c). is the line of application (the “lams rectum'*) in the parametric expression of the 
conics. 

* Apollonius, Conics I, Preface; cf. Pappus. Collection (VII), II. pp. 672 If and 
Heath, Apollonius, pp. xxxi-ii. Futocius misconstrues (his problem as referring to (he 
finding of the two mean proportionals (Apollonius, Opera. II. p. 186). 

**' pur a discussion and reconstruction of the Apollonian investigation of this locus, 
see Heath. Apollonius. Ch. V ami Zeuthen. Ke^ehchnitic. C*h. 7. 8. 

M Pappus. Collection (Vlh. II, pp. 64H IT; discussed hy Heath. Hi story. I, pp. 431 
ff. 

** This follows SimsoiTs account. as reported by I. Thomas, History of Greek Moth * 
ematics, I. pp. 482-484. See also van tier VVaerden. op. cit.. pp. 287-290. 

** For a compact survey of the history of projective geometry, see L. Cremona, 
Elements of Projective Geometty. 1893. pp. v-xii. The classic treatment is by Chaslcs, 
Aper\u historique drs metlunles en geometric. 1837 (IK75; 1889). 

w Van der Wnenlen reconstructs a proof, op. cit.. pp. 287 f. 

M Pappus. Collection (VII). II. p. KH4 (Lemma 12; Lx'inrua 13 treats the nonparalld 
ease). For the entire sci of lemmas, sec ibid.. pp. 866-918. Uriel* accounts appear in 
Heath. History. II. pp. 419-424 and vail der Wacrdcn. Science Awakening, pp. 287- 
290. 

** !ss trois tivre\ drs Porismes d'Ettclidc. I860. The plan of this reconstruction is 
adumbrated in his Apet\u historupte. pp. 274-284. Heath’s assessment of his el loti, in 
light of those of Sintson 11776), Zeuthen 11886). and otheis. appears in History. I. pp. 
431-438. 

* 9 Cf. Heath. History . I. p 437. Heath's principal reservation against accepting 
Chaslcs* reconstruction has hardly any standing at all: it is thal Chaslcs makes out the 
propositions in Fuel id's Par isms to he on a par with Pappus' lemmas, while one should 
expect thal the lemmas were pitched at a far lower level than the theorems in the associated 
treatise. But if such is often the ease with Pappus* lemmas, it seems not always to have 
been so. For instance, (lie neusis problem from Heraclitus (Prop. 72) is surely comparahlc 
to the neusis constructions in Apollonius* work; a hyperbola construction among the 
lemmas to the Conics (Prop. 204) is the effective equivalent of a construction now to be 
found in Conics II. 4 (sec Chapter 7. note 33); the focus-directrix constructions lor (he 
conics (Props. 236-238) must match or surpass their correlates in the Huclidcan Surface 
Loci (sec below). In several instances. Pappus provides problems with full analysis and 
synthesis: Props. 85. 87, 105, 107 -109, 117; and he sometimes offers not only one, hut 
sometimes two alternative proofs of lemmas (c.g.. Props. 35. 36. 39). Hus must surely 
indicate that the lemma had a certain interest in its own right. Finally, one may consider 


the construction of the ellipse conforming to the obliquely viewed circle {Collection VI. 
Prop. 53); this is in fact far beyond the scope of the technical methods employed in the 
associated treatise. Bile I id's Optics. We thus have no reason to be uncomfortable with 
the view that in the instance of an early work like Fuclid’s Par isms. Pappus might provide 
in his lemmas a modification, or even an improvement, on materials in the original. 

** Kcgelschniitc. Ch. 8, especially pp. 173-184; cf. Heath, History. I, pp. 437 f. 
Zeuthen shows how a certain porism enunciated by Pappus can lead to the specification 
of a conic protectively. via its property as a four line locus top. cit.. pp. 169 f; we discuss 
this Iik'iik below). In this w ay. one comes to (lie point of a verification of Pascal's theorem 
on the Itrxagon inscribed in a conic; for that can be seen as a consequence of flic generation 
of a conic as the locus of the vertex of a variable triangle whose other two vertices move 
along given lines, while its sides turn about given points, hi noting this connection with 
the porism. Zeuthen nevertheless acknowledges that the ancients were not likely to have 
recognized the Pascal property, lor Pappus would surely have inentiimed it were it known 
[op. cit.. pp 495 fL 

~ Collection (VII). II. pp 884-886. 

m hi the parallel case, the ratio becomes Zl) : GZ in each instance, as Pappus 
establishes in his Lemma 11. 

M H. S. M. Coxetcr. Projective Geometry. New York, 1964. pp. 38 f; W. T. 
Fishback, Projective and Euclidean Geometry. 2nd ed.. New York, 1969. pp. 67-69; 
I). J. Stmik, Analytic and Projective Geometry. Beading, Mass.. 1953. p. 66; B. \i. 
Meserve, Eundametual Concepts of Geometty. Reading. Mass.. 1955, pp. 61-63. 

99 For instance, the lemma on the ellipse in Col I return VI, Prop. 53 (see note 67 
above) assumes a result proved in Pappus* Lemma 28 to the Portsnts (VII, Prop. 154). 
Heath discusses a remat kable coincidence In'tween Lemma 31 (VII, Prop. 157) and 
Apollonius’ III. 45 on the loci of a central conic {Apollonius, pp xxxix f). 

” Collection (Vll). II. pp. 648 652. 

Cf. Heath. Apollonius . pp. 95 f. whose Icitcnng I adopt in part. 

Cl. thid . pp xxv, xlviii. 

}h Zciillieu does not propose a reconstruction of the three line locus; Hculh develops 
an unnecessarily complicated tiirm based on Cotucs III. 54-56 {thid.. pp. I22f). Thus, 
the present vcision is rny own 

99 lltat is. <2V will bisect all chords parallel to PP’. 

m Tills limitation within the pre-Apollonian reseat dies is stressed by Zeuthen; cf. 
Hculh. op. cit.. pp. Ixxxiv ff, exli 

For details, see Heath, op. t it . pp. 123-125. Although he derives his form of the 
three-line locus from III. 54-56 (see note 76 above), this is immaterial to the subsequent 
derivation of the four-line Incus from the three-line locus. 

*" leltre von den Kegelschnitten. Ch. 7. 8; summarized hy Heath, op. cit.. Ch. v. 

M Cf. Heath, op. cit.. pp. cxlv f. 

** Cf. ibid., pp. cxlvii ix. 

M lltat live points determine a conic is established by Apollonius in Conics IV, 25 
using the harmonic division ol secants drawn to conics; results of the latter type arc likely 
to have been well known in the earlier theory (see below). To effect the locus in the 





-—^ C M 

Figure 25 


manner presented here, we require that if G is on the locus relative to ABCD, then I) is 
on the locus relative to A BCG: that is. AS • DR : DS * DT - AG * GH : GL * GM 
(where DR is parallel to BC, and SDT is parallel and equal to AB). By similar triangles, 
AS : DS - AG : GL, while DR : GH * CR : CG * DT : ST, and ST = GM. Hence, 
as claimed, (AS : DStfDK : DT) * (AG : GLKGH : GM). 

M Here, P is the intersection of the conic with the given diameter TV. 

M Cf. note 76 above; Heath designates his Section Jll, 53**56 by the heading, 4 ‘The 
Locus with respect to 3 Lines Ac” {op. cit., p. 119). Cf. Zeuthen. op. cit.. pp. 132 ff. 
162 f. 

" Cf. Zeuthen. op. at., pp. 122-126, 159 f. 

" T H. Balsam cm these theorems in Apollonius von Perga. p. 134; cf. Zeuthen, op. 
cit. t p. 163: "The ancients knew completely the generation of conic sections via projective 
pencils, apart from their comprehension under the generic concept of projectiviiy.” 

M Strong reservations arc voiced by J L. Coolidge, History of the Conic Sections , 
1945, p. 21: 44 Most surprising is Zculhcn's statement in this connexion (p. 163) that the 

Greeks were familiar with the Chaslcs-Steiucr theorem_l confess that it seems to me 

that there is very little basis for such a general statement. 44 

Note that Bcaugrand. in a letter of 1639 criticizing Desargnes for his none Jussieu I 
approach to the study of conics, secs at once this very application of this Apollonian 
theorem; cf. R, Taton, Desargues. 1951, pp. 187 f. 

Cf. Heath. Ch. iv; Zeuthen. Ch. 15. 

Collection (IV), I, p. 2X4. Ilis first method of triscction employs a neusis. effected 
via a hyperbola {ibid., pp. 272-276), while the second is another solution via hyperbola, 
identical to the third given here {ibid., pp, 282-2X4), On these alternative const met ions, 
sec Chapters 5-7. Arabic and Greek texts related to these items will be presented in the 
sequel to the present volume. 

Hi Collection (VN) Props. 236-238, II, pp. 1004-1014. 

** Pappus gives one other lemma to die Surface Loci t ibid .. Prop. 235, p. 1004): 
the specification of a surface via the condition that its parallel sections arc conics of a 
given relation. The locus reduces either to a cylinder with a conic at its base, or to a 
cone. See Heath's discussion. History, l, p. 440; II, pp. 425 *426; and Archimedes , pp. 


Ixii-v. Note that the text bears this meaning without emendation, despite the feeling by 
Tannery and Heath that there might be some textual problem here. 

^ See Toomer. Diodes. Props. 4. 5 and his discussion, p. 17. The view that Euclid 
knew some form of the focus-directrix property was generally accepted (cf. Heath, 
Apollonius, pp. xxxviii ~ix) even by Toomer himself, until his recent edition of Diodes. 
Bin l oomcr's present view that the appearance of this construction in Diodes must reserve 
to him credit fot the fitst discovery of this property might, I believe, be an overreaction 
beyond what the Diodes text requires. 

For references, sec Heath. History. I. pp. 439 f. 

** Pappus, Collection (IV). I, p, 234, For an account of these passages on the various 
forms of the spirals, sec my *'Archimedes and the Spirals/ 4 

Pappus, op. cit .. pp. 258- 262. Pappus gives another solid construction for the 
quadratrix via the plane spiral; the spiral is related to the conical spiral via orthogonal 
projection, the latter in its turn being projected radially onto the cylindrical spiral. Cf. 
ibid., pp. 262-264, and Chapter 5 below. 

** See Chapter 6 on Nicomcdes and the quadratrix. 

w Note that the focus of the ohliquc parabola will fall at the midpoint of KD. not at 
D. Thus, ndtlicr here nor in the cases of the ellipse anti hyperbola given below will the 
orthogonal projection map a focus onto a focus. 

,,,,, For definitions, see the discussion of F.ratosthcncs 4 loci with respect to means in 
Chapter 6. 

M " Hie extension of this result to any conic is given by Apollonius in Conics HI, 37. 

Ibis is proved by Pappus in Lemma 19 to the Pori sms. 

If the circles |F»g. 2l(a)| are projected onto conics, as fiom the horizontal to the 
oblique sections of cones |Fig. 21<b)|, the polar line (locus of the harmonic division of 
secants) for the former will indeed be projected onto that for the latter. But the two conics 
resulting from the projection of the two circles will not he the locus of the arithmetic 
and geometric means; fot unlike the harmonic division, the proportional divisions cor¬ 
responding to these means are not preserved under projection in general. (Note, for 
instance, that the center of the given circle 0 is not projected onto the midpoint of the 
diameter MS.} A suitable configuration can lie obtained if we introduce instead the sections 
ol a cylinder |l ; ig. 26). Li the given circle PQ and the loci of its thicc means be formed 
as the horizontal section of three intersecting right cylinders, and let the secant KPQ be 
drawn, where Kll, KG, KA are, respectively, the harmonic, geometric, and arithmetic 
means of the segments KP, KQ. If an oblique section of the cylinders is now made, a 
configuration of intersecting ellipses will result, and the points IP. CP, A' will divide 
K'P'Q' into segments in the same proportion as do M, G, A the line KPQ (for the lines 
of projection PP', MU', etc . are parallel). Thus, the line and (he two ellipses which arc 
the respective loci of IP', G\ A' will now be the required loci of the harmonic, geometric, 
and arithmetic means of K'P\ K'Q\ This method of parallel projection will thus gen¬ 
eralize the locus problem for ellipses; but I have not found a comparable procedure which 
might yield the analogous result for parabolas and hyperbolas. Zeuthen takes up these 
locus problems in his effort to reconstruct Eratosthenes 4 study of 4 4 loci with respect to 
means. 4 4 While Zetillicn alludes to the stcrconictrical procedure for ellipses ( Kegelschnitte . 
p. 323), he proposes that Hiatosthencs adopted a planimctric method in order to cover 
(lie eases of parabola and hyperbola as well; cf. the discussion in Chapter 6. 





Q 



Figure 26 

,IU Tlie analogous result for the arithmetic means is trivial, since in this case N is the 
center of the given circle, 

"" As (he points A. A'. P. P’ arc coptanar. (he lines AP. A'P' lie in (lie same plane. 
They cannot be parallel; for then the plane of APN would be parallel to that of A'P'N' 
(since AN, A'N' arc parallel), against the assumption of their intersection along (lie line 
KK . Hence, AP, A P intersect, namely at a point on the intersection of the respective 
planes of the two sections, the line KK’. As noted above, the Desargues theorem in the 
plane is entailed by one of Euclid's porisms; ef. note 64. 

m The exception in the Conics is the sel of problems in I. 52-60 producing the cone 
and its sectioning plane in answer to conics specified via given diameter and latus rectum. 
These, in effect, arc converses of the definitions of the curves established via scctionings 


of llie cone in I. 11-14. Otherwise, the study and application of these curves is always 
effected via planimctric means in the Conics. 

M " See note 47 above. 

For texts and discussions of the work of Desargues and Pascal, see R. Talon, 
L’Oeuvre mailiematupte ,le (,'. Oeungtiex. 1951. An extremely brief, but useful, account 
may he loimd in (oolidge, op. fit ., pp. 28-53. See also the accounts cited in note 63 
above. 

"" Collection (VII). II. pp. 652 654; cf. Heath. History of Greek Mathematics. I 
pp. 432 f. 

"" Collection (VII). H. pp. 676-678. 


I 

l 

I 







CHAPTER 


5 


Archimedes: 
The Perfect 
Eudoxean 
Geometer 

Among the ancient scientists none has attained greater fame, either in his own 
time or in ours, than Archimedes of Syracuse. His immense achievements in 
pure geometry extended into the field of mechanics where his theoretical findings 
had consequences for the design of mechanisms. Siege engines constructed under 
his supervision held the Roman fleet at bay for three years during the ill-fated 
campaign which despite his efforts ended in the city's fall and his own murder 
in 212 B,c In the popular view Archimedes thus became the prototype of the 
scientific wonder-worker* an image sustained by the ancient historians and elab¬ 
orated beyond the limits of credibility by the lcgcndizers. 1 But the Archimedes 
wc meet in his own writings is, with but rare exceptions* exclusively the pure 
geometer. It is his place within the development of the ancient geometry which 
shall concern us here. 

Archimedes was bom near the time that Euclid died* around the second decade 
of the 3rd century B.c. 2 One might thus suppose that Euclid's work was a 
dominant influence in Archimedes’ training as a geometer and that Archimedes 
would eventually enter and extend the field of study opened up by Euclid. But 
this turns out not to be the case. Points of detail relating to Archimedes' early 
studies and his technique of proportions reveal him to depend on a tradition 
alternative to Euclid's Elements * doubtless closer to the manner of the original 
Eudoxean treatments. 3 In the area of mechanics which so absorbed his attention. 
Archimedes could find no precedent in Euclid's work. 4 Yet* as we shall see. the 
area which Euclid did advance, the application of the method of analysis for the 
solution of geometric problems, is one which drew relatively little interest from 




152 Ancient Tradition of Geometric Problems 


Archimedes: The Perfect Eudoxcan Geometer 153 


Archimedes. This split is striking, but not entirely surprising. Archimedes* father 
was an astronomer, hence competent in the Eudoxean geometric methods and 
quite capable of rearing his son on the older geometric treatises. 5 Moreover, the 
separation of Syracuse from Alexandria might well sustain differences in research 
interests. It would thus appear that Archimedes was too close in time, but too 
far in distance to come under the spell of Euclid's particular brand of geometry. 

Characteristic of the Eudoxean geometry is the application of an indirect 
method of limits for the measurement of curvilinear plane and solid figures. The 
centerpiece of this tradition is the issue of the quadrature of the circle. One 
perceives this problem to be an important factor in Archimedes* early geometric 
efforts, particularly the Dimension of the Circle , while it leads naturally into the 
surface and volume measurements carried through in his Sphere and Cylinder I 
and (he studies of areas and tangents in his Spiral Lines.* The same Eudoxean 
approaches find their application in the demonstration of the principles of equi¬ 
librium and center of gravity in Plane Equilibria. From this a powerful heuristic 
method emerges by which the content and centers of gravity of figures can be 
determined through the conceptual weighing of their constituent elements. Ar¬ 
chimedes neatly summarizes the manner of its use in the Method for obtaining 
the results whose formal demonstrations he had already communicated in the 
Quadrature of the Parabola. Conoids and Spheroids , and the lost Equilibria. 1 
These formal proofs adhere meticulously to the style of limits Archimedes de¬ 
veloped in refinement of the Eudoxean method. One thus perceives that the 
techniques and concerns of Eudoxean geometry run through the entire Archi¬ 
medean corpus, as extant in Greek, and indeed define its subject matter at virtually 
every point. 

By comparison, the investigation of geometric problems assumes a low profile 
at best in Archimedes* works. It is the concern of only one of the extant Greek 
writings. Sphere and Cylinder II, where six of the nine propositions arc problems 
seeking the construction of segments of spheres according to specifications of 
size and proportion, and where the double method of analysis and synthesis is 
applied in each instance. Of course, Archimedes shows himself to be complete 
master of the technique. Moreover, works preserved now only through their 
medieval Arabic translations include interesting problem-solving efforts, for in¬ 
stance, an angle trisection implied in one of the propositions in the Book of 
Lemmas and a seven-section of the circle worked out in the Inscription of the 
Regular Heptagon. Both solutions depend on neusis , that is, conceptual manip¬ 
ulation of an idealized sliding ruler, and through this have a natural connection 
with certain constructions, also effected via neuses, introduced in the investi¬ 
gation of the tangents \n Spiral Lines. Few in number perhaps, these constructions 
of problems set a firm precedent for the solution of problems via special curves, 
neuses , and other means by geometers in the following generation. 

Thus, through a survey of Archimedes* contributions to the solution of prob¬ 
lems we cannot hope to reveal the scope, or even capture the essence of his 
geometric achievement. For this must omit discussion of his advanced efforts 
in the measurement of conics and solids of revolution and of his inquiries into 


geometric mechanics and hydrostatics.* But in the context of our other discus¬ 
sions, the work of Archimedes ought to emerge as the product of a powerful 
geometric intellect, at once influential for the development of the field, yet oddly 
out of sympathy with its principal objectives. 

CIRCLE QUADRATURE AND SPIRALS 

No geometric results discovered by Archimedes were more widely known and 
used in antiquity than those derived from his study of the measurement of the 
circle. Later writers cited his Dimension of the Circle for the rule that the area 
of the circle is one-half its perimeter times its radius, analogous to the area rule 
for triangles; they cited it also for approximations: that the circle is n /i4 times 
the square of its diameter, and that its circumference is 3 Vi times its diameter. 
While these three results are established in the three propositions, respectively, 
of the extant Dimension of the Circle , the ancients knew this work in a different, 
more substantial version; for, in particular, they could draw other results from 
it, such as the analogous area rules for sectors and perhaps also an inequality 
appropriate for estimating segments of circles. v Nevertheless, the extant version 
clearly reveals the source of Archimedes’ formal technique and his skill at 
adopting this for computational purposes. 

Most accounts of Archimedes’ works assign this writing to a time relatively 
late in his career. But this view is the consequence of a plain misunderstanding. 
When Heiberg proposed his chronological ordering of the Archimedean writings, 
he saw no clear indications of the placement of the Dimension of the Circle , 
save its obvious affinity in content with the measurements of spheres in Sphere 
and Cylinder I. He thus set it at the end of his list, followed by a question mark 
to indicate his uncertainty. In the frequently consulted survey of Greek mathe¬ 
matics by Heath, this list was reproduced, but without the question mark for 
this entry. 10 Thus, the circle measurement was transformed into a late writing, 
reversing Heiberg's tentative association of it with the relatively early sphere 
measurement. Further consideration supports Heiberg’s intuition, although he 
appears to have underestimated the early dating of the circle measurement. 

What marks the Dimension of the Circle as especially early is its adherence 
to the elementary Eudoxean form of the method of limits. For in his other writings 
Archimedes introduces a refinement on this method in which the curvilinear 
figure to be measured is bounded above and below by figures of known measure; 
as one can arrange that the bounding figures differ from each other by less than 
an arbitrary preassigned quantity, an indirect argument establishes that the in¬ 
termediate figure must equal a specified figure also known to lie between these 
bounds. 11 For instance, Archimedes proves in Sphere and Cylinder I, 5 that to 
a given circle there may be circumscribed and inscribed similar regular polygons 
whose areas have a ratio less than any preassigned ratio greater than unity. It 
follows that such bounding figures may be found to differ by less than a preas¬ 
signed area. To establish the result in Dimension of the Circle, Prop. I, one 
would need only to note that the area determined as one-half the circumference 


154 Ancient Tradition of Geometric Problems 


Archimedes: The Perfect Eudoxean Geometer 155 


times the radius of the circle is greater than any polygon inscribed in the circle* 
but less than any polygon circumscribed about it. For the area of a regular 
polygon is one-half its perimeter times its in-radius; while the perimeters of the 
circumscribed and inscribed polygons are respectively greater than and less than 
that of the circle* their in-radii respectively equal to and less than the radius of 
the circle. Thus, the area of the circle and the product of one-half its perimeter 
times its radius must differ by less than any preassigned quantity. Hence, as 
claimed, they are equal. 12 

Instead of the expected two-sided convergence method, Archimedes follows 
the Eudoxean one-sided method here, in which the lower bounding figures are 
considered entirely separate from the upper bounds. In each instance, doubling 
the number of sides of the polygon reduces at least by one-half the difference 
of its area from that of the circle. 13 Through the same area rules for regular 
polygons stated above, it follows that the circle is neither less than nor greater 
than the specified product; hence it is equal to it. This procedure conforms 
precisely to the Eudoxean form of the circle measurement* as wc know it from 
Euclid's Elements XII, 2, with but one important exception. The Eudoxean 
theorem has to do with the ratios of circles. In the indirect proof, the assumption 
of a lesser inequality can be reduced to a contradiction through consideration of 
inscribed polygons. But instead of introducing the circumscribed figures to elim¬ 
inate the assumption of greater inequality* one may reduce this case to the prior 
case merely by inverting the ratios. This ploy is not available to Archimedes, 
since his theorem asserts an equality of areas rather than of ratios of areas. 
Introducing the circumscribed figures entails a further assumption on the mag¬ 
nitudes of arcs. Just as he had assumed that the perimeter of the inscribed polygon 
is less than that of the circle* so now he must assume that the perimeter of the 
circumscribed polygon is greater than that of the circle. The former assumption 
might be considered obvious by virtue of the conception of the straight line as 
the shortest distance between two given points. This notion was surely familiar 
in the pre-Euclidean period, but it is not articulated as an axiom by Euclid. 14 
To the contrary, an equivalent for the rectilinear case is established as a theorem : 
that in any triangle the combined lengths of any two sides must be greater than 
the third side (I. 20). Later philosophers criticized Euclid for belaboring the 
proof of what is obvious to any ass. 15 But surely Euclid, and doubtless geometers 
before him. intuited the generality of this result. It is hard to suppose, moreover, 
that no attempt was made around the time of Eudoxus to establish the propor¬ 
tionality of circumferences and diameters in view of the analogues for circular 
area and spherical volume in XII, 2 and 18. As far as we know, the requisite 
axioms receive their first explicit formulation by Archimedes among the pos¬ 
tulates prefacing Sphere and Cylinder I: that the shortest distance between two 
points is the straight line joining them; and that for two curves, convex in the 
same direction and joining the same two points, the one which contains the other 
has the greater length. 16 In the Dimension of the Circle , however, one finds no 
specific indication that an axiom of this form underlies the steps that the perimeter 
of the inscribed polygon is less than that of the circle, while the perimeter of 


the circumscribed polygon is greater. Indeed, in this work no justification for 
these steps is given at all. Presumably, then, Archimedes could accept them as 
obvious. In view of his strong dependence here on the earlier techniques of 
convergence, one need not doubt that in these other regards he could find certain 
precedents in attempts by earlier geometers to prove theorems on arc length. 17 
The formulation of these principles as explicit axioms in Sphere and Cylinder 1 
would thus result from Archimedes' own later reflections on the formal require¬ 
ments of such demonstrations. 

The Archimedean area rule in conjunction with the Eudoxean theorem on the 
ratios of circles ( Elements XII. 2) yields the result that the circumferences of 
circles have the same ratio as their diameters. Pappus provides such a proof 
within a series of lemmas on the ratios of circles and their sectors and segments.'" 
and we should expect to find it in the Dimension of the Circle, for Archimedes 
goes on in its third proposition to estimate the ratio of the circumference to the 
diameter, not the ratio of the areas of the circle and the square on its diameter. 
In the extant Prop. 2. the latter ratio is asserted to equal II : 14. but its ‘•proof' 
actually depends on the result of Prop. 3. Clearly, the writing as we know it 
has been altered to its detriment through editorial revisions and scribal confu¬ 
sions.” The lemmas on related properties of the circle which we find in Pappus 
and Hero, for instance, may thus derive from Archimedes’ work before it had 
suffered these textual corruptions. 

Despite such difficulties the essential skill and genius responsible for the 
computations in the third proposition arc unmistakeable. Archimedes seeks to 
establish not merely an approximation for the ratio of circumference to diameter, 
i but upper and lower bounds on that ratio. As in the first proposition, the procedure 

! is divided into two separate sections, one dealing with circumscribed figures 

| (leading to the upper bound), the other with inscribed figures (for the lower 

i bounds). This division masks the fact that the computational procedures arc 

1 identical, each requiring the same property of the bisector of the angle of a 

i triangle (cf. Elements VI, 3). Indeed, complete derivations for both procedures 

I are provided, despite their equivalence. Such intermediate proofs often turn out 

. lo be the work of interpolators. But even if that is the case here, the editor has 

I only adopted and supplemented the separation of cases fixed in the computation 

itself. The failure to exploit the potential for refining this procedure by means 
of the two-sided notion of convergence thus serves to affirm the view that the 
Dimension of the Circle was an early Archimedean effort. 10 

The procedure adopted is as follows. First consider the upper bounds, for 
. which wc take the circumscribed regular hexagon as the initial bounding figure. 

Form the right triangle whose leg A is one-half of one of the sides of this 
hexagon, whose other leg B is the radius of the circle, and whose hypotenuse 
is C. so that the central angle formed by B and C is '/u the full circle [see Fig. 
1(a)). If the corresponding element of the polygon having twice as many sides 
is denoted by A', B’, and C', Archimedes' rule establishes that A' : B' = A : 
B + C, while C' 1 = A' 1 + B ,J . Starting from the hexagon, he can then proceed 
in succession to the polygons of 12. 24. and 48 sides, ending with that of 96 



IS6 


Ancient Tradition of Geometric Problems 


Archimedes: The Perfect Eudoxean Geometer 157 




Figure 1 

sides. In each case, rounding off due to the approximation of irrational square 
roots must give an upper bound on A (that is, a lower bound on B and C), thus 
ensuring that the final value of the ratio of A : B will yield an upper bound for 
tt, namely 96 • 2A : 2B. For the initial value of A : B = I : \/3, he uses the 
upper bound 153 : 265. The manner of its derivation is unexplained, and has 
given rise to many attempts at reconstruction by scholars tantalized by the fact 
that this is one of the convergent fractions in the continued-fraction development 
of the root. 21 After four applications of the rule, Archimedes obtains 153 :4673Vi 
as an upper bound on A: B, from which he produces 96 * 153: 46731/2 = 14688 
: 4673’/: as an upper bound on tt. The final value 3 ‘A follows from this through 
rounding off upward. Here again the effort to find good fractional approximations 
in (he smallest possible terms appears to be assisted by procedures comparable 
to the development in continued fractions. 22 The configuration for the lower 
bounds is altered, in that A is the side of the inscribed polygon, while C—rather 
than 2B—is the diameter of the circle [Fig. I(b)|. Otherwise, the computation 
proceeds as before, where rounding off must always be downward so that the 
ratio A : C will produce 96 A : C as a lower bound on it. The initial value for 
A^B is 780 : 1351 here (another convergent in the continued fraction for I : 

V3).and after four applications of the rule, one obtains 66 : 20I7 1 /* as lower 
bound for A : C. 2 ' Hence, 96 • 66 : 20l7'/4 = 6336 : 20l7'/4 is a lower bound 
for it. This is simplified to 3'%i via another downward rounding off. 24 

In this way Archimedes has established 3'A as an upper bound and 3 K Ai as 
a lower bound on tt. Although the computation requires that the error due to 
rounding off must be cumulative, the final values differ by less than one part in 
2400 and one part in 4200, respectively, from the actual value. The Greek 
tradition is aware of no better estimate before the time of Archimedes, while , 

(he value 3A is employed without exception within the tradition of metrical j 

geometry after him. 25 But closer estimates were derived through further appli- I 

i 

i 

i 

♦ 


cations of Archimedes’ method. Hero reports that Archimedes himself produced 
a different set of bounds in a work On Plinthides and Cylinders . Eutocius cites 
a work called the Easy Delivery (dkytokion) in which Apollonius improved on 
the Archimedean estimates. 26 Eutocius does not elaborate, but Hero does. In the 
alternative estimate Archimedes is said to have established 211875 : 67441 as 
a lower bound and 197888 : 62351 as an upper bound on tt. On first inspection 
these values are disappointing, for the former term is in fact an upper bound on 
it, while the latter, despite the large terms of the ratio, is a far poorer upper 
bound than 3'A. Thus, either Archimedes has blundered woefully in the execution 
of this computation, or else the text has suffered corruption. On the latter hy¬ 
pothesis several attempts at restoration have been made. 27 My own accepts the 
lower bound as given, but proposes that it has been mistakenly cited as a lower 
bound when it was actually derived as an upper bound; for one would not wish 
to suppose that a good value like 211875 :67441 arose merely by chance through 
a clumsily managed computation, while the accuracy is precisely of the expected 
order, about one part of the denominator. 28 Comparable accuracy for the lower 
bound suggests the value 197888 : 62991, where the manuscript reading of 
62351 is easily accountable through scribal error. 29 The correctness of the re¬ 
ceived numerator is indicated by the factorization 197888 = 2 8 * 773, where 
the factors of 2 correspond to the successive doubling of the number of sides of 
the bounding polygons. The ratios as emended can both be obtained through an 
Archimedean procedure commencing with decagons and ending with the 640- 
t gons. But as in the Dimension of the Circle . the computed bounds could be 

J reduced to lower terms with negligible loss of accuracy, yielding 333 : 106 as 

j lower bound and 377 : 120 as upper bound, from the continued fractions 

3 + '47 + Ids) and 3 + x Ai + ’/17), respectively. This at once suggests an 
j alternative intermediate value 355 : 113 = 3 + 'An + 'At). This same value 

■ was established by Chinese geometers in the 5th century a.d.; it was derived 

| again by Western computers in the 16lh century. Since it is accurate to about 

. one part in twelve million, however, the ancient computation implied by Hero’s 

figures could not have established it rigorously save through some essential 
modifications in procedure.' 0 

Since the value 3 'A suffices for the practical purposes of metrical geometry, 
ancient documentary evidence of such more refined computations is sparse. Only 
i in astronomy was a closer estimate demanded. For the trigonometric studies by 

Hipparchus around the mid-2nd century B.C., Toomer has reconstructed the value 
« 3438 : 21600 as effective estimate for the ratio I : 2 tt. 31 Since this might be 

derived simply as the arithmetic mean of the inverses of Archimedes’ values in 
the Dimension of the Circle , we cannot cite it in support of Hero’s testimony to 
the refined Archimedean values. Indeed, the larger terms of the latter indicate 
! a degree of accuracy about ten times that required for Hipparchus’ uses. In the 

J great system of mathematical astronomy compiled by Ptolemy three centuries 

later the value adopted for tt is 3 P 8' 30" in the sexagesimal notation, that is, 
377 /i 2 o. 32 This is precisely the value we derived from the upper bound reported 
by Hero. Ptolemy provides no derivation of the value, but observes merely that 


158 Ancient Tradition of Geometric Problems 


Archimedes: The Perfect Eudoxcan Geometer 159 


it lies between the Archimedean values 3*96i and 3 l A. Although he might well 
have depended on refined computations such as that cited by Hero, an alternative 
account is equally possible on the basis of the trigonometric table in Ptolemy s 
Book I. There the value for chord 1° is l p 2' 50* for a circle of diameter 120*; 
one might thus produce for it the estimate 360 • chord 1° : 120 = 3 P 8' 30" : 
l Ptolemy sets out in full detail the computation leading to his value for chord 
1°, and both this value and the estimate for it corresponding to it arc correct to 
the nearest digit in the second sexagesimal place after the units. Thus, his results 
do not of themselves indicate prior knowledge of this computed estimate for -it. 
This does not rule out that such knowledge might have helped steer the course 
of the computation of the chords. For instance, the value of chord 1° is found 
via the inequality chord l 1 /* 0 < chord 1° < 4 A chord W*, where successive 
bisections of the angle 24° have led to the results chord 1 Vi° = l p 34' 15" and 
chord W =0 P 47' 8". This leads to chord l p = ! p 2' 50", but only because 
Ptolemy has chosen to adopt an upward rounding off for chord %° (= (P 47' 
T 24'"...) instead of the expected downward rounding off. Had he instead 
adopted the lower figure, the value for chord 1° would become l p 2’ 49" and 
give rise to a lower estimate for it (e.g., 3 P 8' 27"). These difficulties are resolved 
by carrying through the computations to the third place after the units, as the 
commentator Theon shows. M But awareness in advance that 377 : 120 was a 
good approximation for -it could have served as a reliable guide. 

The juxtaposition of Archimedes and Ptolemy is interesting in that it reveals 
the difference between the theoretical and the practical approaches to this com¬ 
pulation. Ptolemy seeks good approximations for the practical needs of calcu¬ 
lation. Expressing results to the nearest digit gives him the advantage of cancellation 
of errors, so that he obtains values rather closer than those attainable via a strict 
Archimedean procedure. Now his value 3 P 8' 30" turns out to be slightly greater 
than tt, despite its association with the table of chords, that is, a derivation based 
on inscribed polygons. The Archimedean procedure, by contrast, controls the 
direction of rounding off, so that only lower bounds can result from the inscribed 
figures. Moreover, the production of both upper and lower bounds provides a 
direct indication of the degree of accuracy of the approximation. In this way, 
Archimedes reveals a greater concern for the theory of this computation than 
for the particular numbers it happens to yield. Thus, one cannot be satisfied by 
the view, already voiced in antiquity, that Archimedes' aim in this work was 
essentially practical.” This might indeed characterize the older Mesopotamian 
efforts, doubtless dependent on empirical measurements, which resulted in the 
value 3 , /k. v * But it fails entirely to capture the subtlety of Archimedes’ effort. 
His value of Vh is certainly more accurate than the older figure. But how is it 
more ‘‘practical” when one considers the manipulation of fractions in ordinary 
contexts? Furthermore, what is “practical” about his lower bound 3 ,( tti? If, 
then, Archimedes’ purpose in the Dimension of the Circle was to deploy geo¬ 
metric theory for the presentation of an algorithm and then to illustrate it through 
a cleverly managed calculation, he would in effect be demonstrating the greater 
effectiveness of theoretical over purely practical procedures in geometry. But 


this makes puzzling the motive behind the extended computation implied in the 
figures reported by Hero. For there is little if any theoretical insight to be gained 
merely by canying out the same computational procedure to one or two more 
decimal orders of accuracy. Neither would this be expected to produce results 
of greater utility for practice. One thus has some cause to question Hero’s 
assignment of these numbers to Archimedes, and to suspect that perhaps another 
geometer, like Apollonius in his Okytokion, was responsible for them. We shall 
return to this question later.” 

The results in the Dimension of the Circle cannot be viewed as attempts at a 
direct solution of the circle quadrature. But ancient and medieval commentators 
recognized that the proof of the area rule in the first proposition depends on the 
postulate that one can produce a straight line equal to the circumference of the 
circle.' 1 * In effect, then, Archimedes has not solved the circle quadrature, but 
rather revealed its equivalence to the problem of producing that straight line. 
Many of Archimedes’ theorems have the same force: they demonstrate that the 
measurement of a specified curvilinear figure can be reduced to that of a certain 
circle. For instance, in the version of the Dimension of the Circle known to 
Hero, Archimedes proves that the area of a sector is one-half the product of its 
radius times its arc. w Presumably, the same work included the proof that the 
surface of a truncated right cone equals one-half its slant height limes the cir¬ 
cumference of its base, as one may see by rolling it out into a plane circular 
sector. In Sphere and Cylinder I, 14 the same result is expressed in the form 
that the conical surface equals the circle whose radius is (he mean proportional 
between the slant height and the radius of the base circle. Similarly, the principal 
result of that work proves that the surface of the sphere equals four times its 
greatest circle, while the area of any segment equals the circle whose radius 
equals the line drawn from the vertex of the segment to a point on its circular 
base. 40 In like manner, the volumes of conical and spherical segments are referred 
to the volumes of cylinders. In Conoids and Spheroids (he area of the ellipse is 
shown to be equal to that of (he circle whose diameter is the mean proportional 
between its major and minor axes; the volumes of conoids of revolution are 
expressed in terms of associated cones. 41 In all these instances, then, the mea¬ 
surement of the figure has been reduced to the fundamental result of the mea¬ 
surement of the circle. 

Two such quadratures appear in the Book of Lemmas . a work preserved in 
Arabic and whose translator Thabit ibn Qurra reports opinions of its Archimedean 
provenance. 42 In the fourth proposition the figure “which Archimedes calls 
arbelos (leather cutter) is drawn, a trigram bounded by three semicircles tangent 
at the points A, D, C (Fig. 2(a)). If BD is the perpendicular half-chord in (he 
large semicircle; then the area of the circle of diameter BD equals the arbelos 
ADCBA. The proof substitutes the relation BD 2 = AD • DC into 
AC 2 o AD 2 + DC 2 + 2 AD • DC to obtain BD 2 = 14 AC 2 - '/* AD 2 - 'A 
DC 2 ; since circles have the ratio of the squares on their diameters, this establishes 
that the circle of diameter BD equals the difference between the semicircle of 
diameter AC and the two semicircles of diameters AD. DC. respectively, where 


160 Ancient Tradition of Geometric Problems 




the latter difference is the arbelos. A similar quadrature appears in proposition 
14: ihe salmon (salt cellar) 4 ' is bounded by the semicircle ABG, the two equal 
semicircles on the diameters AC and BD. and the semicircle DFC [see Fig. 
2(b)). By a proof like that for the arbelos , it is shown to equal the circle of 
diameter GF. One observes that the proofs for both figures are very much in the 
style of the lunule quadratures by Hippocrates of Chios. But there is a forward- 
looking aspect implicit in two additional lemmas to the figure of the arbelos . In 
Prop. 5 it is proved that the circles EFG, LMN are equal, where each is tangent 
to the large semicircle, to one or the other of the small semicircles, and to the 
line which separates them [see Fig. 3(a)]. In Prop. 6 the diameter EF of the 
circle drawn tangent to all three semicircles is diameter AC when AD : 
DC = 3 : 2; that is. CP : PO : OA = 4 : 6 : 9 [see Fig. 3(b)]. 44 The latter 
result finds its natural extension in the "ancient proposition** which Pappus 
presents in Book IV of the Collection: if a succession of such tangent circles is 
inscribed in the arbelos . the elevation of the center of the first above the base 
diameter equals its own diameter, the elevation of the center of the second equals 
twice its own diameter, the elevation of the center of the third equals three times 
its own diameter, and so on |see Fig. 3(c)]. 4 ' The theorem given by Pappus may 
in its tum be viewed as an application of special cases of the problem of con¬ 
structing circles tangent to three given circles, the complete solution for which 


Archimedes: The Perfect Eudoxean Geometer 161 



(c) 

Figure 3 


was worked out by Apollonius in his work On Tangerines. In this way one can 
perceive a link between studies initiated by Archimedes and an inquiry pursued 
later by Apollonius. We shall later discuss comparable links between their studies 
of circle measurement and neuses . Unfortunately, the confused state of the 
manuscript tradition of the Lemmas available to the Arabic translator prevents 
our delineating precisely those pans actually due to Archimedes. The quadratures 
of the two figures are examples, if quite simple ones, of the class of curvilinear 
measurements which dominate the known work of Archimedes. But the theorems 
on the tangent circles fall within a field of problems not otherwise represented 
in the Archimedean corpus, but quite characteristic of efforts in the generation 
after him. We might thus well prefer to assign these results to a follower of 
Archimedes, rather than to Archimedes himself. 

The determination of the areas of figures bounded by spirals further illustrates 
Archimedes* methods of quadrature. The Archimedean plane spiral is traced out 
by a point moving uniformly along a line as that line rotates uniformly about 
one of its endpoints. The latter portion of the treatise On Spiral Lines is devoted 
to the proof that the area under the segment of the spiral equals one-third the 
corresponding circular sector. For instance, the area lying between the spiral 
and the radius vector BE is one-third the sector BEH; the area between the whole 
spiral and the terminal radius BA is one-third the whole circle of radius BA (see 
Fig. 4). The proofs are managed in full formal detail in accordance with the 
indirect method of limits. The spirals are bounded above and below by sum¬ 
mations of narrow sectors converging to the same limit of one-third the entire 
enclosing sector, for the sectors follow the progression of the square integers. 46 
This method remains standard to this very day for the evaluation of definite 
integrals as the limits of summations. It is applied throughout the Conoids and 
Spheroids as well for the measurement of the volumes of solids of revolution. 



162 Ancient Tradition of Geometric Problems 


Archimedes: The Perfect Eudoxean Geometer 163 





Bui Pappus reports an alternative treat mem. "a remarkable strategy” {thaumaste 
epihole), by which Archimedes effected the quadrature of the spiral. 4 ’ As (his 
occurs in the context of communications between Archimedes and Conon of 
Samos, the latter having been deceased for ”many years” at the time of the 
writing of the extant Spiral Lines . Pappus here must be preserving for us a much 
earlier treatment of this theorem. As in the later method. Archimedes here 
introduces a set of inscribed sectors bounding the spiral from below (see Fig. 
Ma)|. At the same time he considers the parallel slices in a cone, (hat is. the 
thin cylinders inscribed in it. whose sum bounds the cone from below (see Fig 
5<b)|. Since in both instances the constituent elements follow the progression of 
t c square integers, it follows that their sums bear the same ratio to their respective 
containing figures. Thus, the lower bounding figure for the spiral has the same 
ratio to the whole circle that the lower bounding figure for the cone has to the 
cylinder. As a comparable proportion obtains for the circumscribed bounding 
figures, it follows that the spiral is to its circle as the cone is to its cylinder, 
namely, one-third. By a closely related technique. Pappus later establishes that 
the spherical spiral separates the surface of the hemisphere into two portions 




equal, respectively, to eight times the segment ABG and eight times the triangle 
ADG, where the sector ADGB is a quadrant of a great circle in the sphere (sec 
Fig. 6). 4 " As for the plane spiral, lhe surface of the figure bounded by the 
spherical spiral toward the initial position ONK is approximated above and below 
by surface elements corresponding to the elements in a second diagram, namely, 
the small sectors drawn to the segment ABG. This depends on Archimedes' 
theorem from Sphere and Cylinder 1 that the area of the surface element 6N0 is 
proportional to the square of 00. just as the area of the sector BHG is proportional 
to the square of BG, where BG = BO. 4 * Hence, the sums have the same ratio 
to their respective enclosing figures. It follows (details of the limiting argument 
arc omitted by Pappus) that the figure bounded by the spiral is to the hemisphere 
as the segment ABG is to the sector AGZ. The sectors AGZ and ABGD equal 
each other, however, while the latter is one-eighth the surface of the hemi¬ 
sphere.* 1 Thus, as claimed, the figure bounded by the spiral is eight times the 
segment ABG. Taking the differences, we have that the other portion of the 
hemisphere bounded by the spiral equals eight times triangle ADG. The extremely 
close similarity in method used for evaluating the area of this form of the spiral 
to that of the plane spiral (as in Pappus) and its heavy dependence on results 
from Archimedes’ Sphere and Cylinder encourage assigning it an Archimedean 
provenance. One may note that the spherical spiral is a striking example of a 
curvilinear figure, like the lunules of Hippocrates, found to be equal to a con¬ 
structed rectilinear figure. 

It is Archimedes’ results on the tangents to the spirals which move toward 
the actual solution of the circle quadrature. Given the spiral whose initial ray is 
OB. the tangent drawn to any point C on the curve meets the line drawn from 
0 perpendicular to the ray 0C such that the intercept 0E equals the arc CF in the 
circle of radius 0C which subtends the angle COB (Fig. 7). Archimedes devotes 
the major portion of Spiral Lines to the formal demonstration of this result for 
the three cases of arcs equal to, less than, or greater than the whole circumference 
of the circle (Props. 18. 19, and 20. respectively), together with the requisite 
lemmas on the general properties of the curve and its tangents. The treatment 
is entirely formal in the synthetic manner, so that the underlying conception of 





164 Ancient Tradition of Geometric Problems 


Archimedes: The Perfect Eudoxcan Geometer 165 




the argument is rather difficult to discern. I thus propose to reconstruct an 
analysis, closely modelled on the synthesis Archimedes gives, to reveal how its 
essential features emerged. One should then be able to work through the details 
of Archimedes* formal treatment with greater ease.' 1 

We may frame this analysis as a problem: to draw the tangent to a point on 
the spiral, for example, the point completing its first full turn. This will be done 
by specifying the length of the subtangent OE where the tangent drawn to B 
meets the line drawn perpendicular to the ray OB (Fig. 8). Let a point Q be 
taken on the spiral on the forward side of B and let the ray OQ be drawn, meeting 
the circle of radius OB at P. By the definition of the spiral, the rays arc in 
proportion to the corresponding angles, that is, to the arcs of this circle; thus 
OQ : OB s arc PB + C : C, where C equals the circumference of the circle of 
radius OB. Since OB = OP, we obtain OQ — OB : OB = QP : OB = arc PB : 
C. Now extend OQ to meet the tangent at F, and draw the line BP extended to 
meet in H the line drawn from 0 parallel to the tangent. Draw from B the line 
parallel to OE meeting OH extended in T. This makes OEBT a parallelogram, so 


that OE = BT. Now by similar triangles, FP : BP = OP : PH = OB : PH. Since 
F is the point on the tangent corresponding to Q, FP > QP ; since, also, PB < 
arc PB, it follows that FP : BP > QP : arc BP. From the above, QP : arc BP 
= OB : C. Thus, OB : PH > OB : C, or PH is less than C. Furthermore, PH 
is less than BT, since BT subtends the largest angle in the triangle BHT. Since 
BT = OE, it follows that PH is less than both OE and C. This does not establish 
a relation between OE and C as such. But it is evident that as P tends to B, Q 
and F converge, so that FP : PB tends toward equality with QP: arc PB; that is, 
PH converges toward C from below. At the same time PH tends toward equality 
with BT. We thus have that the subtangent OE equals C, the circumference of 
the circle of radius OB, the common limiting values of the line PH. One might 
investigate in comparable fashion the relations arising for P situated on the 
rearward side of B. Moreover, the result generalizes for portions other than the 
full turn of the spiral. When the terminal ray OC docs not coincide with the 
initial ray OB, the subtangent OE will be the length of the arc corresponding to 
the sector angle BOC in the circle of radius OC, where OE is perpendicular to 
OC (cf. Fig. 7). 

Clearly, a treatment of this son lacks the precision demanded of a formal 
exposition. But it reveals the length of the subtangent one requires for drawing 
the tangent, as well as the general line of the argument one might follow to 
obtain the formal synthesis. In particular, one perceives the role of the insertion 
(ncusis) of a length shorter than the subtangent, that is, the line PH between the 
circle and the line OH parallel to the tangent.' 2 One might thus try to show that 
any line other than BE as drawn here will meet the spiral at another point besides 
B; that is. that if OD is less than OE, the line BD will cut the spiral at some 
point R on the rearward side of B, while if OD* is greater than OE, the line BD' 
will cut the spiral at Q on the forward side of B. This will establish that BE is 
the tangent, in accordance with the conception of the tangent in the ancient 
geometry.'*' Alternatively, one might consider the hypothesis that the tangent 
meets the perpendicular at E such that OE is not equal to C. The case where OE 
is greater than C (« OG) gives rise to the diagram we have been considering; 
for any point D lying between E and G, one can insert the length PH - OD and 
so locate a point Q on the forward side of the spiral from B, where Q will be 
found to lie beyond F, the point on the extension of the assumed tangent EB; 
this contradicts the fact (proved in Prop. 13) that Q must lie below the tangent. 
Similarly, one may derive a contradiction from the assumption that the subtangent 
is less than C. This approach to the synthesis is in fact the one adopted by 
Archimedes in Spiral Lines, Prop. 18. 54 

The informal analysis proposed here utilizes a somewhat dynamic conception 
of the limiting process in which as P approaches B, the triangle FPB becomes 
similar to BOE, while triangle OPH becomes congruent with OBT. To be sure, 
such conceptions are alien to the formal style of the ancient geometry. Yet one 
would not like to deny to Archimedes the availability of intuitions of this sort, 
especially in view of the importance of mechanical heuristic procedures for his 
other work. Moreover, it is difficult to imagine how one would approach the 




166 Ancient Tradition of Geometric Problems 


Archimedes: The Perfect Eudoxean Geometer 167 



problem of the tangents to motion-generated curves like the spiral without some 
essentially kinematic intuition. One may note that the present form partly dis¬ 
guises the role of the differentia) triangle as one would introduce it in a more 
modem treatment. The triangle FPB would assume that position in the limit, 
where its side FP will be proportional to the radial velocity of the point tracing 
out the spiral as it arrives at B, while BP will be proportional to its velocity 
along the tangent to the circle at B (Fig. 9). Since the motions are uniform, the 
velocities will be proportional to the total distances traversed by B in these two 
senses, so that FP : PB = OB : C. But by similar triangles, FP : PB = OB : 
OE. Hence, OE = C. The procedure actually adopted by Archimedes is far 
removed from this simpler method, so that one must suppose that its conception 
of the tangent as the direction of the instantaneous motion composed from the 
motions which generate the curve, while fundamental for studies of tangents 
since the 17th century, was still unrecognized by the ancients. 55 

The plane spiral is but one of several forms of spiral studied by the ancient 
geometers. We have already mentioned the spiral on the sphere, whose quadrature 
paralleled that of the plane spiral in Pappus* account. Pappus also reports of a 
construction relating the plane spiral to the spirals drawn to the cone and the 
cylinder. 56 Given the plane spiral BHA, if one erects KH = HB, perpendicular 
to the plane of the spiral, the point K will lie on the right cone of vertex B and 
axis BL (Fig. 10). Thus, as H traces the plane spiral, K will trace a “surface 
locus,** the spiral on the cone. 57 If the ray LK is now extended to meet at 0 the 
right cylindrical surface drawn over the circular base GDA, this point 0 will 
trace another surface locus, the spiral on the cylinder. The latter was familiar 
to the ancients by the name cochlias. Apollonius devoted a writing to its study, 
while Archimedes is said to have invented the water-lifting device which shares 
both its name and its figure. 511 We should thus suppose that the ancient geometers 
discovered a variety of properties of these curves. It happens that the projective 
relationship just cited provides an alternative way to find the tangent to the plane 
spiral. First consider the cylindrical spiral: since the ratio 0D : arc DG is a 



constant for all points e, the curve may be viewed as the result of wrapping a 
plane triangle GAM against the cylinder.' 1 '' so that the tangent to 0 will be the 
line 0E where DE is perpendicular to DB and equal to the arc DG. Since the 
cylindrical spiral is the radial projection of the conical spiral, the tangent to the 
latter at K will project onto the tangent to the former at 0 via lines radiating 
from LB. and hence onto the line KZ. meeting the base plane in Z. such that 
DE : HZ = OL : KL. HZ will equal the arc of the circle of radius KL corre¬ 
sponding to the arc DG (= DE) of the circle of radius BD. Now the tangent to 
the conical spiral at K lies in the plane tangent to the cone at K. that is. to the 
plane through BK and perpendicular to the plane of BLK. This tangent plane 
meets the base plane in the line BP perpendicular to BH. while the tangent line 
to the conical spiral will meet BP in N, where ZN is parallel to HB. Since 
HZ = BN. BN will equal the arc of the circle of radius BH corresponding to 
the angle GBH. The plane spiral is the plane projection of the conical spiral, so 
that its tangent at H is the plane projection of the line KN, that is. the line HN. 
We may thus conclude that the tangent to the plane spiral at H meets the line 
BP drawn perpendicular to the ray BH such that the subtangent BN equals the 
arc of the circle of radius BH over the angle GBH. In this way, the conception 
of the spirals in the manner of surface loci has the potential to provide a heuristic 
for the results on the tangents to the plane spiral presented by Archimedes in 

Spiral Lines. ' 

Did Archimedes view his findings on the spirals as a construction of the circle 
quadrature? A sign that he did seems to lie in his manner of expressing them: 
they are not given as problems in the drawing of tangents to spirals, but rather 


168 Ancient Tradition of Geometric Problems 


Archimedes: The Perfect Eudoxcan Geometer 169 


as theorems on the properties of the tangents. Moreover, the later tradition did 
indeed include Archimedes' spirals, along with the quadratrix and the cylindrical 
spiral. among the known efforts at circle quadrature “ What. then, of its success? 
Certainly his followers did not consider the matter closed, for the studies of the 
quadratrix. for instance, appear to have been stimulated by Archimedes’ efforts, 
not cut off by them. As for Archimedes’ view, a remark from the preface to his 
Quadrature of the Parabola is suggestive: 

Now of those who earlier applied themselves in geometry, some tried to prove 
\graphein: confirm by diagram| that it is possible to find a rectilinear space which 
is equal to the given circle and to the given segment of a circle, and after these 
things they attempted to square the space bounded by the section of the whole 
cone and a line by making assumptions not easily conceded, whence it was not 
acknowledged by the majority that these things had been discovered | heuriskomena] 
by them. But we understand that none of the earlier ones has tried to square the 
segment bounded by a line and a section of the right-angled cone (sc. a parabolic 
segmentj. which is what has now been discovered by us.*' 

At first it would seem unclear why Archimedes should view the successful 
quadrature of the parabolic segment as a distinctive achievement within this 
scries of studies of curvilinear areas. For, after all. the lunulcs of Hippocrates 
and his own measurement of the spherical spiral, if the above view of its Ar¬ 
chimedean origin and dating is accepted, would be familiar as instances of 
curvilinear figures found to be equal to certain rectilinear figures. Of the study 
of the ellipse (if that is what the obscure phrase ’’section of the whole cone” 
refers to) 6 “ we know nothing other than the Archimedean quadrature given in 
Conoids and Spheroids . Still, there would appear to be a natural connection with 
the problem of circle quadrature. In this respect, the allusion to studies of 
segments of circles is of interest. For Hero preserves the full demonstration of 
the theorem that the segment of the circle is greater than four-thirds the triangle 
inscribed in it, that is, having the same base and altitude as the segment. 6 ' The 
treatment is in the same manner of Archimedes* Dimension of the Circle , and 
depends on the fact that the triangles AZB, BHG are together greater than one- 
fourth triangle ABG, where Z, H bisect the arcs AB, BG, respectively (Fig. 

11(a)]; applying this result, in turn, to the triangles inscribed in the remaining 
segments of the circle, one obtains that the segment is greater than triangle ABG 
plus its fourth plus the fourth of its fourth and so on, where the sequence of 
parts converges to four-thirds. If one adapts this very procedure for a parabolic 
arc ABG and takes E, K to be the bisectors of the lines AD, DG, then one finds 
that the triangles AZB, BHG are together exactly one-fourth the triangle ABG, 
so that the summation yields that the parabolic segment is exactly four-thirds 
triangle ABG IFig. 11(b)]. This is the manner of Archimedes* proof of this 
theorem in Quadrature of the Parabola , Props. 18-24. M Appreciating how an 
approximate quadrature of the circular segment so gives rise to an exact quad¬ 
rature of the parabolic segment, Archimedes might well feel encouraged in the 
search for a circle quadrature. 


B 



B 



Here Archimedes says nothing about the spirals. But the view that he had 
already made his basic discoveries relating to these curves is supported by internal 
evidence. In the preface to Spiral Lines , communicated several years after Quad¬ 
rature of the Parabola , Archimedes speaks of a list of theorems he had sent off 
to Conon years earlier. The promise to provide their formal proofs was being 
fulfilled in a writing already sent ( Sphere and Cylinder II), together with the 
present work ( Spiral Lines) and another yet to follow ( Conoids and Spheroids). 
He states the theorems at issue, and among them are the principal results on the 
areas and the tangents of the spirals. Now Conon had died many years before 
the writing of the treatise On Spiral Lines ; indeed, his recent death had been the 
occasion for Archimedes to address the Quadrature of the Parabola to the new 
correspondent Dositheus. 65 Thus, before this Archimedes had communicated to 
Conon some form of these results on spirals, doubtless in a manner like that 
preserved by Pappus. 

It thus appears that Archimedes’ solution of the circle quadrature via the 
tangents to the spiral might fall into that class of efforts “making assumptions 
not easily conceded,** so that the problem remained open. This lends a certain 
poignance to a remark he makes in the preface to one of his latest writings. The 
Method , addressed to Eratosthenes: 

It happens that these theorems Ion the volumes of certain sections of cylinders) 
differ from those discovered earlier Ion the volumes of conics of revolution). For 
the latter figures, namely the conoids and the spheroids and their segments, were 
compared by us in magnitude to figures of cones and cylinders, but none of them 
was discovered to be equal to a solid figure bounded by planes. But each of those 
figures (newly communicated] bounded by two planes and surfaces of cylinders is 
discovered to equal one of those solid figures bounded by planes. 




170 Ancient Tradition of Geometric Problems 


Archimedes: The Perfect Eudoxean Geometer 


171 



t 

* 

i: 


Despite an entire career of diligent and masterly effort, the quest for measure* 
ments of curvilinear plane and solid figures equal to rectilinear figures had 
produced but a handful of actual instances, while the circle quadrature itself 
remained as elusive as ever. 

PROBLEM SOLVING VIA CONIC SECTIONS 

In the extant Archimedean corpus, only Sphere and Cylinder II includes appli¬ 
cations of the method of analysis and synthesis. It is devoted to the solution of 
problems dealing with spherical segments: c.g., the division of the sphere into 
segments whose surface or volume are in a given ratio; or the construction of 
segments equal to a given figure and similar to a second. These depend on the 
principal results of Sphere and Cylinder I: that the surface of the segment ABG 
(excluding its base) equals the circle of radius AB; and that the volume of the 
sector equals the cone whose base equals the surface of the segment and whose 
altitude equals the radius of the sphere (Fig. I2)/* 7 'Hie form of the latter result 
brings to mind Archimedes’ expression for the area of the circle in Dimension 
of the Circle , as equal to the triangle whose base is the circumference of the 
circle and whose altitude is its radius. Indeed, Archimedes himself notes this 
analogy elsewhere as significant lor his discovery of the measurement of the 
sphere.** 

It is quite straightforward, then, to divide a given sphere into segments whose 
surfaces have a given ratio (Sphere and Cylinder II, 3). Since these segments 
have the ratio AB 2 : AE 2 = BX : XE, one need only divide the diameter of the 
sphere into segments having the given ratio and then divide the sphere into parts 
having these line segments as their respective altitudes. The problem of dividing 
the sphere into segments where the volumes are in the given ratio, however, 
turns out to be a major undertaking (Prop. 4). Merc Archimedes makes use of 
an expression for the volume of the segment established in Prop. 2: if on the 
diameter extended, one marks off the point V such that Kl) I l)X : DX = PX: 
XB (for K the center and DB the diameter of the sphere) (Fig. 13), then the 
cone whose base is the circle of diameter AG and whose altitude is PX will 
equal the segment of (he same base with vertex at B. If one now determines the 
analogous point L producing the cone l.AG equal to the segment DAG, then 
the analysis of the problem of dividing the sphere reduces to that of determining 
the three points X, P, L such that (a) PX : XI.- is the given ratio; for the cones 
are as their altitudes and by construction they arc equal, respectively, to the 
segments of the sphere; (b) KD + DX : DX = PX : XB; and (c) KB + BX : 
BX = LX : XD, from the determinations of P, L as just stated. Through a 
considerable exercise in the manipulation of proportions, Archimedes reduces 
this in its turn to the problem of dividing 111 ) at X such that BD 2 : DX 2 = XZ:Z0, 
where ZB = BK and UZ : Z0 = PL : LX. Since PX : XL is given and BK is 
the radius of the sphere, Z and 0 arc given, so that only the position of X remains 
to be solved with reference to the derived condition. One perceives that it entails 
the solution of a third-order relation in DX 3BK- ZX). For the present 



Archimedes merely assumes the solution of this problem for the purposes of 
producing the synthesis of the problem of dividing the sphere, but promises that 
‘'this shall be analyzed and synthesized at the end/'** 

Through some textual mishap suffered by the work not long alter its compo¬ 
sition, (he construction of this auxiliary problem became dislodged from the rest. 
Already by the dose of the 3rd century n c , the geometers Diodes and Dion- 
ysodorus could find no such construction in the copies of Sphere and Cylinder 
available to them for study, and so each worked out his own form of the solution. 
These arc reproduced by Eutocius. who reports that his own researches among 
the library shelves had brought to light an old hook in deplorable condition which 
presented a treatment of these mailers. 7 " Since it was written in the Doric dialect, 
and as it referred to (he conic sections hy their archaic names ("section of the 
right-angled cone." rather than "parabola”), he deduced that this was an Ar¬ 
chimedean work, for both features are characteristic of his writings. Eutocius 
goes on to present this text in a restored version devoid of archaisms and replete 
with appropriate citations of Apollonius' Conics. To this he adds versions of the 
methods used by Dionysodonis and by Diodes. We possess an alternative text 
of the latter, to which Eutocius’ version is reasonably faithful (the principal 
discrepancy being that Eutocius presents not only the analysis, but also the 
synthesis, while Diodes omits the latter as "obvious”). 71 Thus, only the hy¬ 
pothesis of outright forgery on Eutocius' part could lead us to question that his 
version of the Archimedean method represents reliably the one originally prom¬ 
ised hy Archimedes. Not only arc the motives for such a forgery entirely unclear, 
but also there is good cause to doubt that the expertise of a commentator like 
Eutocius was such as to produce an original treatment of comparable insight, 
yet so different from the alternatives which might serve for models. 72 




172 Ancient Tradition of Geometric Problems 



Figure 14 


Let us sketch the analysis of Archimedes* solution of this problem. We arc 
to divide the given lines BD at X such that BD 2 : DX 2 = XZ : Z0 for given 
points Z and 0. But it is now posed in the more general form in which the given 
area E may be any value (not necessarily BD 2 ), while the given line segment F 
is likewise arbitrary (not necessarily Z0). Assume then, that the given line ZD 
(= ViBO) has been cut at X such that ZX : F - E : XD 2 (Fig. 14). Draw ZO 
equal to F at right angles to ZD and complete the parallelogram 0IUR with 
diagonal 0XJ. We now determine M on 0H extended by the relation E =0H • HM. 
Thus, (a) XZ : Z0 = OH • HM : XD 2 by the relation for X. while (b) XZ : 
Z0 = 0H : HJ by similar triangles, so that (c) XD 2 = HM ' HJ. If now we 
draw Ql through X parallel to HJ. since Ql = HJ and QJ = XD, it follows 
that Q lies on a parabola with verlex at H, axis HJ and latus rectum HM (i.c., 
QJ 2 = HM • HJ). Furthermore, from (b) Ql • 10 (= HJ • XZ) = DH • HO 
(= Z0 • HO), so that Q lies on the hyperbola passing through D with asymptotes 
HO, 0Z. Since the parabola and the hyperbola arc given, Q is given as their 
point of intersection. 

This completes the analysis of the problem: the synthesis can be given merely 
by constructing the conics to determine Q and using this to establish that X docs 
indeed divide ZD as to satisfy the desired relation. Now it is obvious that the 
original problem of dividing the sphere into segments of a given ratio is always 
capable of solution, for one or the other of the segments may take on values 
ranging from arbitrarily small to arbitrarily close to the whole sphere. Thus, to 
every ratio, however small or great, there will be a corresponding division of 
the sphere. But the auxiliary problem considered by Archimedes lias been gen¬ 
eralized, so that it is no longer obvious that a solution will exist for certain 
choices of E and F, the given area and the given line, respectively, ’litis is 
apparent from the construction derived in the analysis, for one is not necessarily 
assured that the conics derived will indeed intersect within tlic specified range 
over ZD. Hence, a tliorism, or determination of the conditions of solvability. 


Archimedes: The Perfect liudoxean Geometer 173 



becomes necessary. Archimedes provides only the synthetic proof of the con¬ 
dition to l>c fulfilled: that the quantity E • F be less than ZY * YD 2 , where 
2 ZY = YD. But it is not difficult to perceive the analysis which leads to this 
condition: the limit of solvability occurs where the hyperbola and the parabola 
have a common tangent at their intersection Q (Fig. IS). In the case of the 
parabola, this entails that JH =IIN, N taken as the point where the tangent 
meets the axis Jll extended 7 ’; hence, QJ = 2HS (for S the point where the 
tangent meets (III), so that DY = 2SI. As for the hyperbola, the point of tangency 
bisects (he segment of the tangent falling between the asymptotes, so that 
TQ = QS. Thus 01 = IS, whence DY = 2YZ. M Although Archimedes em¬ 
ploys the tangents in the synthesis of (his condition, his aim there is to prove 
that of all divisions X of the line DZ, this one yields a maximum for the value 
of ZX • XD 2 . It thus follows that if E • F is greater than ZY • YD 2 , the stipulated 
division of ZD will be impossible. 

Archimedes’ treatment of this problem is a nice example of how the process 
of generalizing stimulates the field of geometric studies hy raising new questions 
for solution not posed in the special problems initially. We shall sec several 
instances of this in the work of Apollonius, especially in his studies of neusex 
and plane loci. Moreover, his manner of investigating the normal lines to the 
conic sections is rcmarkahly close to the Archimedean problem we have been 





174 Ancient Tradinon of Geometric Problems 

discussing here, including the use of intersecting conics to determine critical 
points, the properties of tangents to discover the limits of solvability, and even 
their elaboration in terms of maxima and minima. 75 Nevertheless, certain po¬ 
tentials in Archimedes* results here appear not to have been recognized for further 
development. Consider that Archimedes has shown how to draw conics to de¬ 
termine X such that ZX : I 7 = If : XD* for any given line l : , area K, and line 
ZD, where ZX + XD = ZD. Thai is, he has constructed the solution for a 
general class of cubic relations and determined the condition under which a 
solution exists for X within a given range. One might formulate other expressions 
of this type, ultimately to obtain an exhaustive classification of cubic relations 
together with the construction of their solutions under specified conditions of 
solvability. This is the very enterprise which Omar Khayyam undertakes in his 
algebra, and the method of solution he uses depends on intersecting conics. 76 
As Eutocius* commentary on the Sphere and Cylinder was familiar among the 
Arabic geometers, wc need not doubt that Archimedes* method in this problem 
could serve as a model for these studies. But if comparable efforts to generalize 
and classify these cubic constructions were taken up in antiquity, the extant 
record preserves no direct indication to that effect. 

Indeed, without the model of Archimedes* solution to work from, Diony- 
sodorus and Diodes prove quite insensitive to these possibilities of generaliza¬ 
tion. Their efforts to find solutions to the problem assumed without proof in 
their copies of Archimedes remain strictly inside the coniines of the problem of 
dividing the sphere. Eutocius inaccurately describes Dioriysodorus as unable to 
cope with Archimedes* form of the auxiliary problem, so that he was forced to 
derive an alternative reduction starting from the terms of the initial problem.' 7 
This characterizes the method adopted by Diodes, but Dionysodorus actually 
follows the line of Archimedes’ treatment quite closely. Although Eutocius 
reports only the synthesis, the analysis underlying Dionysodorus* procedure is 
not difficult to perceive. Assume that the given line ZD has been divided at X 
such that XZ : ZO = BD 7 ; !)X\ where Zfl and BD arc also given (Fig. 16). 



Figure 16 


Archimedes: The Perfect Eudoxean Geometer 175 

Since XZ : ZO = XZ • ZO : Z0\ it we set (a) QX 2 = XZ • ZO, it follows that 
QX J : Z0 J - BD J : DX\ (hat is. (b) QX : ZO « BD: DX. The locus of Q that 
answers (a) is a parabola with vertex Z, axis ZD. and latus rectum ZO; that which 
answers (b) is a hyperbola with asymptotes ZD and the line perpendicular to it 
through D. such that the curve passes through the point V for BV - ZO. Hence, 
; the intersection of these two conics determines that point Q lending to the required 

division of ZD at X. No dioristn is provided; hut as we have already noted, 
none is needed, since the restricted problem of division is always solvable. 
Although the actual curves derived by Dionysodorus differ from those introduced 
by Archimedes, 7K the close resemblance of the two methods to one another cannot 
be missed. 

By contrast. Diodes returns to the initial conditions of the sphere division, 
I specifying the points X, L. and P via three proportionalities. Here I present only 

the construction he derives in the analysis: assume as known the positions of L, 
X. and P in relation to the diameter DB of the sphere (Fig. 17).* Draw Dd at 
right angles to DB at D and of given length (namely, in the context of the 
restricted problem, one-half the diameter DB; but here Diodes allows for any 
value of the given line). Complete the rectangle defg with diagonal dXf, draw 
{ QXli parallel to cf. and extend eX to meet dg extended at j. Next determine k 

and m such that Dk ~ Bm = Dd. the given length, and draw through them 
lines perpendicular to DB extended to meet at r and s. respectively, the line 







176 Ancient Tradition ot Geometric Problems 


Archimedes: I lie Pcrlcct hmloxcun Geometer 177 


which makes half a right angle with DB at B. Let the line rBs met QXh at n. 
One finds that (a) Qn 2 : m • ns = PX : 2 XL, where the ratio PX : XL is given; 
and (b) Qh • hd = Be • ed. The former specifies Qn as the ordinate of an ellipse 
of diameter rs and whose latus rectum has to rs the ratio PX : 2 XL.* 1 The latter 
specifies Qh as the ordinate of a hyperbola passing through B and having ed, 
dg as asymptotes. Since these curves arc given, Q will be found as their inter¬ 
section, and from 0 one has the required point X dividing DB. Diodes presents 
only the analysis of the problem; the synthetic demonstration is added by Hutocius. 

Diodes, like Dionysodorus, holds the narrow context of the problem of 
dividing the sphere, and so misses the opportunity for constructing the more 
general problem with its diorism in the manner of Archimedes. Yet wc do well 
to note the essential similarities uniting these three presumably independent 
approaches. Doubtless, the procedure of applying intersecting conics for the 
solution of problems was relatively undeveloped at Archimedes* time, although 
Menaechmus* cube duplication was of course an established precedent for it. 
Neither Dionysodorus nor Diodes can refer to Archimedes* solution, while 
Diodes at least appears to come before Apollonius wrote the Conics. Their 
efforts thus reveal the maturity not only of the theory of the conics nl the time 
near or before Apollonius, but also of the field of problem solving in which the 
conics were applied. 

Another problem of this “solid** class, that is, entailing the use of conic 
sections for its solution, arises in connection with Archimedes* theorems on the 
tangents in Spiral Lines. As we have seen, the proofs of the determinations of 
the subtangents in Props. 18-20 assume certain points Q on the curve, and these 
are found via points P lying on the associated circle. The latter refer hack to 
lemmas (Props. 5-9) in which P is found by inserting a given length PH between 
the circle and a given line. But how is this neusis to Ik effected? Apparently, 
Archimedes accepted it as obvious, by virtue of the very terms in which the 
operation is introduced; that is. it is the abstraction of what one may easily 
visualize in the form of the manipulation of a sliding ruler. But later geometers 
were not comfortable with this assumption cm his part. Pappus reports a criticism 
to this effect: 

It seems somehow to he no small error among geometers, whenever one solves a 
plane problem via conics or |higher| lines, and in general, whenever a problem is 
resolved [via curvcs| from a class not akin [to that of the problem); for instance, 
the problem of the parabola in (he filth book of the Conics of Apollonius and the 
neusis of a solid toward a circle which is assumed by Archimedes in the book on 
the spiral; for without the intervention of any solid it is possible to find the theorem 
provided by him |sc. Spiral Lines. Prop. I8|."' 

One would suppose that the objection relates to the use of solid methods, that 
is, conics, for solving problems where the more elementary planar methods 
suffice. That is indeed the case for the Apollonian passage. 112 In the Archimedean 
instance, however, the alternative constructions of the neusis via planar tech¬ 
niques either appeal to nonconstructive assumptions or require major modifi¬ 


cations in the structure of Archimedes* proof.*' Moreover, when Pappus goes 
on actually to “set out the analysis of the neusis assumed by Archimedes in the 
book on spirals, so that you might not be at a loss,** the method he uses is 
“solid,** for it requires conic sections.* 4 While the text is in some disorder, it 
has Ikch restored to make sense. It is in the form of two lemmas on the con¬ 
struction of the conics, leading to the solution of the neusis ; in each case only 
an analysis of the construction, without the synthesis, is given. 

The following paraphrase will show its relation both to Spiral Lines and to 
the solving methods used for the solid problem of Sphere and Cylinder II. It is 
required to insert a given length between a given circle and a given line as to 
pass through a given point on the circle.* 5 Here the line is a diameter LN of the 
given circle of center 0 and radius OR; let the line PH he inserted between them, 
where PH has the given length 0 and its extension passes through the given point 
B on the circle (Fig. 18). Since HB and HL are secants to the circle, 
NH • ML = Pll •III!. If we draw KM perpendicular to LH such that (a) 
KH = HB, it follows that (b) Nil • UL - PH * KM; since PH is given, the 
locus of K is a parabola, for if V is the intercept with the diameter RUR\ then 
OV • I) = NO *01. ()N 2 , while Kll • 0 = Nil • (NH -F 20N), so that 
0 • (KH 4 0V) = 0 • VW = OH 2 = KW 2 ; that is. K is on the parabola of 
vertex V and axis VO and latus rectum (I. Furthermore, (a) sets K on a hyperbola 
of vertex B and diameter BB' ( = 2BZ) such that KY 2 ( = HZ 2 ) = BY • YB\ 
The intersection of these two conics thus produces the point K from which the 
required position of Pll follows. 

While this construction docs not directly require the “intervention of a solid,** 
as Pappus oddly expresses it, this doubtless is the “solid neusis" at the base of 
his objection to Archimedes* procedure." 7 We must then suppose yet another 







178 Ancient Tradition of Geometric Problems 

treatment which somehow dispensed with the use or conics. This seems to be 
indicated in Pappus* remark following upon this construction: 

Now some accuse |Archimedes| of having needlessly employed a solid prob¬ 
lem;. . .they show how it is possible also to find by means of planes the line equal 
to the circumference of the circle, applying the cited theorems on the spiral. M 

But now an even graver confusion has set in. The unnamed critics seem to 
suppose that their alternative to the neusis converts the construction into a planar 
solution of (he circle quadrature, despite its dependence on the spiral itself, that 
is, a “linear** method, as well as on the drawing of its tangents. Clearly, the 
latter cannot be effected without introducing the subtangent, and hence requires 
a prior solution of the circle quadrature. Thus, neither Pappus nor his source 
betrays a clear perception of the nature of the problem, (hat is, under what 
conditions a proposed circle quadrature will be judged proper."'* 

As for this method of effecting the neusis via conics, Archimedes was perfectly 
able to produce such a construction. But that he did not is clear from Pappus* 
need to supply it, filling a gap in the Archimedean presentation. Furthermore, 
Archimedes could hardly have expected his readers to assume such a construction 
of the neusis as obvious. It thus appears that he would not have admitted the 
force of the objection Pappus raises against the introduction of neuses. We shall 
see additional instances of this difference in attitude, as wc turn next to other 
problem-solving efforts by Archimedes. 

PROBLEM SOLVING VIA NEUSES 

Arabic sources preserve for us two Archimedean problems which, like the prob¬ 
lems in Spiral Lines , rely on neuses for their solution where alternative procedures 
making use of conics can be found, atul indeed were found by later geometers. 
One of these is an angle triscclion implied by a proposition in the Hook of 
Lemmas , the work wc consulted earlier for Archimedes* quadratures of the 
arbelos and the salinon figures. The second effects the inscription of a regular 
heptagon in a given circle. In both cases we arc indebted to Thahit ibn Qurra 
for the Arabic translations which preserved knowledge of these results and stim¬ 
ulated a host of related efforts among Arabic geometers. 

The writing On the Inscription of the Regular Heptagon consists of 17 prop¬ 
ositions, most devoted to properties of triangles unrelated to the problem of the 
title.'" 1 In its last two propositions one finds first a lemma establishing certain 
properties of a figure constructed by means of a neusis, and then the application 
of this figure for solving (he problem of inscribing the heptagon. The procedure 
is synthetic throughout, so that the underlying line of thought tends to be ob¬ 
scured. It is of course natural to suppose that the original investigation started 
off with the analysis of the problem,'" and on this view Tropfke presented a 
partial reconstruction. Wc shall follow his lead as far as he was able to go, and 
then introduce an auxiliary analysis to complete that part of the argument which 
he admits baffled him. 


Aiclumcilcs. 1 lie IVilal l.mnuctiii «•i.nintu.* 


1 1 • 


H _ L 



E 

Figure 19 


Let it be assumed that the heptagon has been inscribed in the given circle 
(Fig. 19), so (hat the side llll is the chord of an arc one-seventh of the circle, 
the diagonal AM is the chord of the double arc HLA. and the diagonals AB, 
RZ. UK are chords of triple arcs; let AB, HZ meet in C and HE, ZB meet in T 
and join CT. In the triangle 1)1 IB. the angles at II anil B arc equal, since they 
subtend the equal arcs BE. 11 A; thus lines III) and DB arc equal. In the triangles 
IIDC, BUT, the vertical angles at D arc equal, while the angles at II and B are 
equal, since their associated arcs EZ, ZA arc equal; thus the triangles arc similar, 
and since HI) = DB, they are also congruent. Hence. HC = TB. The quad¬ 
rilateral 11 BTC thus has equal sides HC. TB and equal base angles at II and B; 
it may then be inscribed in a circle, so that angles CTH, CBH arc equal (for 
they arc subtended by the same chord CIO. Now angle CBH is twice CAM; 
while DO! equals the sum of CHA, CAII, that is, twice CAH. Thus angle CBH 
equals each of ihc angles DCH, CTH, This makes the triangles CTH. DCH 
similar, so that 111 • HD » IIC 1 . Since Til * CB.HD = DB.andHC = CA, 
one has (a) CB • BD « AC\ a relation of the segments of line AB. Furthermore, 
since CHD and DAI I arc equal angles, triangles CHD and DAH arc similar, so 
that AD * DC* = Dll'. Since HD = DB, it follows that (b) AD • DC = DB J . 
a second relation of the segments of AB. Thus the problem of inscribing the 
heptagon has reduced to that of finding two points C, D dividing the line AB 
according to the relations (a) and (b).*** 

This derived problem is solved in Archimedes’ preliminary lemma. It is there 
shown that if on the given line CB, one draws Ihc square CBFG with diagonal 
UG. and the neusis of line AF is effected so that the triangles KFU, LCA arc 
equal in area, then the segments of BA will satisfy the relations (a) and (b) (Fig. 
20a). Again, only a synthesis is given. But one can construct an analysis which 
leads from these relations to the condition of the neusis in a reasonably straight¬ 
forward manner.'*' Let it be assumed that the line AB has been divided at C and 






338 Ancient Tradition of Geometric Problems 


correspondence in content linking the ancient and modem theories of conics is due in no 
small part to the intense diligence with which 16th- and 17th-century geometers studied 
the works of Apollonius and Pappus. 

^ See above, notes 24 and 25. 

Sec Chapter 4. I have collected and discussed the testiiitonia on Aristacus in my 
article, "On the Early History of the Conics." 

** Collection (VII), II, p. 672 

Hypsieles in the preface to Book XIV of the Elements. Sec also Heath, Euclid's 
Elements . Ill, p. 512; and History, l, pp. 419 f. One may note that of the traditional 
fifteen books of the Elements . only the first thirteen arc actually by Euclid. The fourteenth 
is by Hypsieles; the fifteenth is of uncertain authorship, but parts of it are by a 6th*ccntury 
disciple of Isidore of Miletus. 

See K. Vogel, "Aristacus" in the Dictionary of Scientific Biography. In the article 
cited in note 95 I suggest that this second Aristacus may have been the father of Hypsieles, 
for their relation to the effort by Apollonius on the regular solids would appear to be of 
the same sort. 

** See notes 26 and 33. 

Pappus. Collection (IV), I. pp. 282-284. Hie construction is paraphrased by Heath, 
who mistakenly makes the latus rectum \/3 AH instead of its triple {History, J, pp. 241 — 
243). The correct reading is adopted by Htiltsch and Ver Eeeke. 

,1M See Chapter 4. 

M,? Collection (Vll), II. pp. 1004-1014. 

On the early expressions for the conics, sec Chapter 4 and note 63. 

HM Collection (Vll). II. p. 672. 

Ibid., (IV). I. pp. 298-302. See also my article. *’Archimedes' Neusis -Construc¬ 
tions in Spiral I Joes.” and Chapter 5tii) above. 

See note 75 above. 

,,,T See note 39 above. 

"** Dozens of times throughout his commentary on the Conics , Eutocius explicitly 
cites the "scholia in the copies" on which he depends for alternative proofs and other 
explicatory material. One thus has little doubt that Pappus, too. depended on such manu¬ 
scripts with annotations for the lemmas lie provides to the Conics and other analytic 
works in Collection Vll. Furthermore, one would expect that annotations in a work like 
the Solid Uni would reflect the significance of Apollonius* form of the theory of conics 
among later geometers. Thus, a source of this nature could welt account for the major 
portion of the solid problems presented by Pappus. 

,w I present in the sequel the texts, with discussion, of the methods of akSijzf and 
al-QuIu. 

I elaborate the view that Anthemius’ study is somehow based on pre-Dioclean 
sources in the article cited in note 80 above. 

See note 91. 

nt Collection (VI), II, Props. 53-54. We have mentioned this section in connection 
with Euclid’s Porisms in Chapter 4. 

,M Conics, preface to Hook I fed. Heiberg, I, p. 4). 


CHAPTER 8 


Appraisal 
of the 
Analytic Field 
in Antiquity 

ll is generally supposed that after the time of Apollonius the study of geometry 
entered a period of decline, interrupted only briefly by the work of Pappus In 
the 4th century a.d. This view agrees with the single most obvious and significant 
fact that the later tradition of geometry never again witnessed the rise of a first- 
rate mathematical mind comparable to Archimedes or Apollonius. On other 
counts, however, the view is questionable. The assumption, for instance, that 
"Pappus stands out as an accomplished and versatile mathematician" has never 
been subjected to scrutiny and is not likely to survive it. 1 More importantly, lltc 
view implicitly assumes that the later figures sought to make original contri¬ 
butions, Init merely failed through technical incompetence. Until we obtain a 
better sense of the methods and motives of these figures, we shall thus be barred 
from an adequate assessment of their achievement in relation to that of their 
precursors. 

We may usefully distinguish three aspects of their efforts: technical, textual, 
and evaluative. The technical competence of later writers like Hero, Thcon, and 
Eutocius to expound mathematical texts, add minor lemmas or appropriate ci¬ 
tations to the standard textbooks, and so on is manifest throughout their works; 
equally clear is (lie absence of original findings on their part, or even the intent 
to produce such. By contrast, the work of Pappus, in particular his Collection . 
contains such a rich diversity of more advanced materials that one naturally tends 
to assume his originality, save where he explicitly acknowledges his dependence 
on a prior source. The rigorous testing of this hypothesis would entail a detailed 
examination of all parts of the Collection , and (his evidently overreaches the 
scope of the present context. A start is made in the sequel with the discussion 


vi Kvia'm.i- ■ 1 





■ <Ti ™ J T t- *« r*Tf ATVV, •>*^ VV r < x? ^r- 


340 Ancient Tradition of Geometric Problems 

of his treatments of the geometric problems, and this reveals that his effort is 
not essentially different from that of any other commentator in this later tradition. 
If Pappus might on occasion submit a result as a finding of his own, as in the 
case of his method of cube duplication, we can perceive the clear dependence 
on earlier methods which effectively undercuts his claim. His extensive review 
of the works on analysis in Book VII only rarely rises above the level of what 
one might expect to find in the margins of annotated copies of these treatises, 
precisely the source exploited by Eutocius for the production of his commentary 
on Apollonius. 3 The more interesting parts of Book VII usually bear signs of a 
provenance in sources; for instance, the enunciation of the role for measuring 
solids via centers of gravity is related to the theorems “in the 12th book of these 
Elements,** and so appears to betray its placement within a commentary on 
Euclid and, as a comparison with a passage from Hero has indicated, its orig¬ 
ination with Dionysodorus late in the 3rd century b.c. 5 It is especially significant 
that the entire analytic corpus surveyed in Book VII derives from 3rd-century 
| geometers, primarily Apollonius and Euclid. Surely one must infer that the 
t problem-solving activity embodied in those works received no appreciable ad- 
f vancement at the hands of later geometers. 

The decreased attention to advanced geometry beginning in the 2nd century 
; B.c. should not be taken as a decline of the mathematical field in general, but 
rather a shift of interest from the special field of problem solving toward others, 
especially spherics, trigonometiy, and numerical methods, which bore directly 
on the development of geometric and computational astronomy. Such shifts in 
the fashion of research, as it were, are a recurrent phenomenon in the history 
of science and mathematics, and indeed, changes of this sort had happened at 
least twice before within the ancient geometry. The study of irrational lines, for 
instance, was a prominent concern among geometers in the pre-Euclidean period; 
it engaged the efforts of superior geometers like Theaetctus and Eudoxus, while 
the formal synthesis of the classification of the irrationals constitutes the largest 
and most tightly structured part of Euclid's Elements * Yet wc learn of no work 
in this field among later geometers beyond a study of "unnrdered irrationals" 
by Apollonius and a scattering of isolated results in Pappus which lie entirely 
within the Euclidean framework. 5 Similarly, Eudoxus* limiting methods were 
the chief inspiration for the work of Archimedes and of necessity remain an 
element in the study of the circle quadrature and related problems of curvilinear 
measurement. Nevertheless, their significance to geometers other than Ar¬ 
chimedes was minimal, as one appreciates most strikingly in their utter absence 
from the works of Apollonius. 6 Doubtless, in the cases both of the study of 
UTationals and of the application of the Eudoxean techniques, the prospect of 
new and interesting discoveries was discouraging, so that later geometers largely 
abandoned their study for other fields. In much the same way, the field of analysis 
must have made the impression of having reached saturation through the accom¬ 
plishments of Apollonius and his colleagues. 

Thus, the later writers on geometry are not likely to be of great interest in 
connection with the development of technical methods or the discovery of new 


. Appraisal of the Analytic Field in Antiquity 341 

results. They are extremely important, however, with respect to the issue of the 
textual transmission of these technical materials. The editorial tradition of col¬ 
lecting and commenting on interesting geometric results extends at least as far 
j back as Hero, Geminus, and Menelaus in the 1st century a.O. and continues 

j almost without interruption through late antiquity into the circle of Arabic schol- 

) ars in the 9th and 10th centuries. Thcon and Eutocius contributed critically to 

• the preparation of editions of the works of Euclid, Archimedes, and Apollonius. 7 

It is thus through these commentators that we receive the overwhelming majority 

; of our data relating to the ancient geometry. Understanding their editorial ob¬ 

jectives and methods is thus indispensable not only for a view of geometric 
I activity in later antiquity, but for one of the earlier researches as well. The 

detailed comparison of alternative texts which such an inquiry demands will be 
I undertaken in the sequel to the present volume. 

A third aspect of potential interest in these later writings lies in their efforts 
to evaluate the geometric achievement of their own and earlier times. The com¬ 
mentators typically volunteer observations on general matters dealing with the 
structure and methods of the field. In effect, they preserve a shadow of ancient 
metamathcmatical thinking, touching on such issues as the classification of prob¬ 
lems, the relative status of the different constructing methods, the nature of the 

* distinction between theorems and problems, and the role of the method of anal¬ 
ysis. From their remarks we may hope to determine their positions on two 

» questions hearing on the special problems of construction which have been our 

continuing interest in the present study: What precisely were the conditions to 
be imposed on a proper solution of these problems? Did the ancient search for 
solutions attain its goal? As we turn now to a survey of these matters, wc may 
j anticipate that the ancients* assessments differed in significant ways from those 

one would expect in a modem account of geometric constructions. 

THE ANCIENT CLASSIFICATIONS OF PROBLEMS 

In two passages preliminary to separate discussions of the special problems, the 
one (A) devoted to the cube duplication, the other ( B ) to the angle trisection, 
; Pappus sets out in virtually identical terms the classification of geometric prob- 

1 lems which one now generally accepts as the standard view among the ancients:* 

The ancients : Wc) say lhai there are three kinds of problems in geometry: some 
of them arc called "planar,** others "solid,** and others "linear.** Now those 
which arc capable of being solved [lyein) by means of straight line and circular 
arc would quite reasonably be called "planar**; for the lines by means of which 
such problems are found {heurixkein) have their genesis in a plane. But those 
v problems which arc solved when there is assumed toward their discovery (heuresis) 

one or several of the sections of the cone are called “solid"; for their construction 
(kataskeui) necessarily employs surfaces of solid figures, namely Ihe conic sur¬ 
faces. Yet a third kind of problems is left, the one called "linear"; for there are 
taken for their construction lines different from those just mentioned, having a 
more diverse and rather contrived genesis (R adds: being generated from less regular 
surfaces and intertwined motions. Such are the lines found in the so-called “loci 


t 





342 Ancient Tradition of Geometric Problems 


Appraisal of the Analytic Field in Antiquity 343 


with respect to surfaces'* 9 ... and those found from the intertwining of plectoid 
surfaces and all manner of other surfaces and having many wondrous ( thaumasta ) 
properties about them). Such happen to be (B\ Of the same kind arc) the spirals 
and quadratriccs and conchoids and cissoids (A adds: having many marvelous 
< paradoxu ) properties about them). 1,1 Since there is such a differentiation of the 
problems, the ancient (H\ former) geometers were not able to construct (ft: lind) 
the above-mentioned problem of the two mean lines (B\ of the angle) when they 
adhered to the geometric manner (/oiar) (B: when they sought it by means of the 
planes), for it is by nature solid. 11 

Pappus recognizes an exactly analogous division of loci, as he explains in a 
remark prefacing his commentary on Apollonius' Plane Loci : 

Those loci which concern us here are called "planar" and in general arc such as 
arc straight lines and curves (!) or circles. Uut solid loci are such as are sections 
of cones—parabolas or ellipses or hyperbolas. Loci arc called "linear** such as 
arc lines other than straight lines or circles or any of the cited conic sections. 11 

This classification scheme is of course not original to Pappus. He himself ascribes 
it to "the ancients," while his designation of the problem of cube duplication 
as "solid" is confirmed much earlier by Hero.' 5 As for the loci, Apollonius 
composed a work On Planar Loci which does indeed examine locus problems 
solvable via straight lines or circles. If (he Aristacus who compiled (he treatise 
On Solid Loci was in fact a contemporary of Apollonius, as I have argued in 
Chapter 7, then this work would surely have dealt with those loci solvable as 
conics; for Apollonius describes the third book of his own Conics as containing 
"many remarkable theorems useful for the syntheses of the solid loci." 14 

Commenting on the last-mentioned passage, however, Eutocius presents a 
somewhat different view of the division of the loci: 

It wns customary for the ancient geometers to speak of "planar loci" whenever 
in the case of problems the problem comes about not only from a single point, but 
from several. 11 ... Such as these arc called "planar loci." but the loci called 
"solid" take their name from the fact that the lines by means of which the problems 
corresponding to these loci are drawn have their genesis from the section of solids, 
in the manner of the sections of the cone and many others. There are also others, 
called "loci with respect to a surface," which have their name from the special 
property relating to them.'* 

In this instance, Eutocius is clearly drawing from a different body of information 
and perhaps inferior to that accessible to Pappus two centuries earlier. Yet he 
appears to maintain a more primitive view than Pappus, one bringing together 
alt cases generated from solids, rather than differentiating the "solid" from the 
"linear" type. Both writers call attention to the "loci with respect to surfaces;" 
for Pappus these fall within the "linear" category, while for Eutocius they would 
appear to constitute a class of their own. Now, Euclid wrote a work On Surface 
Loci , and it was seen in Chapter 4 to be connected not only with certain locus 
problems involving conics, but also with certain space curves, including forms 
of spirals, and the plecioidal curve which projects onto the quadratrix in the 


plane. One would thus suppose that the wide variety of cases introduced in this 
Euclidean work came to be viewed as a class of loci additional to those loci 
connected with the constructions in the Elements f and only later was this new 
class reconsidered within the tripartite system known to Pappus. The same thing 
appears to have happened lo the "loci with respect to means" arising from the 
work of Eratosthenes discussed in Chapter 6; for Pappus himself treats these 
loci as a class separate from his basic three, yet ease by ease assignable to one 
or other of them. 17 

Surprisingly, Proclus more resembles Eutocius than Pappus in his view of 
the division of loci: 

I designate as "theorems of locus" those in which the same property obtains over 
some entire locus, ami as "locus" ilofws) the placement (thesis) of a line or a 
surface which makes one and the same property. Vur of the locus theorems some 
arc framed with respect to lines, others with respect to surfaces. And since some 
lines arc planar, others solid—planar arc those whose conception in a plane is 
simple, such as that of the straight line; while solid are those whose genesis arises 
from some section of a solid figure, such as of the cylindrical spiral and (he conic 
sections— I would then say that of the locus theorems with respect to lines, some 
have a planar locus, others a solid locus. 14 

In admitting both conics and spirals into his "solid” category, Proclus agrees 
with Eutocius rather than with Pappus. Yet in speaking of these as forms of 
"line locus" lie differs from both, since they would term these "surface loci.” 
Proclus goes on to report, on the authority of Geminus, a view of Chrysippus: 
that he likened the class of locus theorems to the "ideas” by virtue of their 
bringing together an infinity of eases answering to the specified terms. 19 It is 
reasonable lo infer (hat the more llcxible classification of loci shared by Proclus 
and Eutocius also has standing in the older sources, and that the tighter view 
proposed by Pappus need not express the consensus of the ancient geometric 
field. 

What of the provenance of Pappus' view? Clearly it would be impossible 
before ihe 3rd century D.C., since distinguishing the “solid” class must indicate 
a substantial development of the theory of the conic sections. One may detect 
from Pappus’ passage on the loci that he is not reporting a very ancient view 
there, for he inserts it to elucidate a distinction of quite a different sort drawn 
from Apollonius' Planar Loci.™ Apollonius is said to denote loci as "epheclic,” 
"diexodic” and "anastrophic,” according to whether the locus of a point, line, 
etc. is respectively of its same dimension or one or two dimensions higher. 21 
Pappus then indicates that planar, solid, and linear loci in his own sense can 
have one or another of these designations. It would appear that in his work 
restricted to planar loci, Apollonius did not have to remark on the special nature 
of other kinds of locus, like the solid class. This suggests that Pappus’ long 
accounts of ihe division of the problems (passages A and B above) might include 
his own editorial elaborations of notions only partly prefigured in his sources. 22 
Indeed, one may well suspect that his rationale for their naming lacks any real 
authority from the older tradition, for it is essentially incoherent. How can one 




344 Ancient Tradition of Geometric Problems 1 

I 

presume to account for the "planar** class through its dependence on lines which j 

are formed in a plane* when the same is no less true of the conic lines, the plane \ 

spiral, the quadrat rix, the conchoid, and so on? Similarly, the "solid” class 

employs only the sections of the cone, indeed only its planar sections, rather 

than the full range of planar and space curves formed by the sectioning of the 

cylinder, cone, sphere, torus, and other solids. 21 Pappus himself elsewhere insists 

that the solid class is not to be understood through such a reference to solids, J 

but only to the conic sections. 24 Discrepancies like this indicate that neither j 

Pappus nor his sources had a clear rationalization for these designations. ! 

If Pappus' view is thus a faithful reflection of the original sense of this 
classification, we would have to suppose that at the time when the field of 
problems had been sufficiently advanced so that geometers wished to distinguish j 

between those solvable by means of straight lines and circles and those solved 
by means of conic sections, they seized upon the salient difference in the manner 
of generating these curves, the fonner lying in the plane, the latter arising through 
the sectioning of solids, and so named the respective classes "planar” and 
“solid,** without considering the inconsistencies entailed. Extant evidence can¬ 
not deny the possibility of this view. But one can hardly be blamed for seeking 
a more satisfying alternative. Now, the terms "planar locus” and “solid locus,” 
in the sense used by Pappus, are attested with Apollonius, but not earlier. 2 ' If 
they were coined by him, or very near his time, it is surely possible that his 
reasons for the choice were more profound than those just given. Now, in the 
case of problems, whenever a line segment is to be found in answer to a second- ! 

order relation, the construction can be effected by means of circles and straight 
lines, as one knows from the older techniques of the application of areas. Fur¬ 
thermore, when the segment is specified via a third-order relation, its construction * 

may be effected via the intersection of conics; we have seen an important example I 

of this in Archimedes* solution of the division of the sphere, while the cube : 

duplication of Menaechmus is an older and more obvious instance. M Working 
out the details of this claim relating to the solid constructions would demand 
only the investigation of a number of additional cases, the type of inquiry j 

characteristic of the treatises in the Euclidean and Apollonian tradition of anal- j 

ysis. Under this alternative view, it would be an intuition of the algebraic structure I 

of the problems, as it were, which motivated the selection of terminology. Many 
of the cases taken up in Apollonius* Plane Uhi seem quite remote front geometric 
interest, yet are easily appreciated as resulting through the urge to set out the 
variety of forms which give rise to solving lines and circles. 22 This view need 
hardly press us to suppose that the ancients attained, or even sought, a complete 
reduction of the field of problems to such algebraic forms. But it provides a 
sense in which an ostensibly arbitrary classification of problems, one whose 
rationale evaded the later commentators, might have originally been based on a j 

real insight into the nature of these questions. 2 " j 

Terms like Euclid's “locus with respect to surfaces** and “locus with respect \ 

to three and four lines** and Eratosthenes* ”locus with respect to means.** for 
instance, are purely descriptive in their intent. In particular, they distinguish 


Appraisal of the Analytic Field in Antiquity 343 

certain classes of problems front others by virtue of the conditions which define 
the loci, rather than by considerations of the methods used in their construction. 
The terms “planar locus** and "solid locus** thus differ from the others in that 
they do seek a distinction on the basis of the construction method. Even so, 
their introduction need not at lirst have been intended for other than descriptive 
purjHiscK. With Pappus, however, the distinction obtains a normative aspect: 

The following sort of thing somehow appears to be no small error to geometers: 
whenever a planar problem is (omul by someone via conic or linear |lincs|. and 
on the whole whenever |some problem) is solved from a class other than its own. 
for instance, the problem on the parabola in the fifth hook of Apollonius* Conics 
and the solid nettsis toward a circle assumed by Archimedes in the book On the 
Spiral , for by using no solid one is able to find the theorem proved by him,* 

’litis rather timid pronouncement is the only ancient statement I know of which 
articulates the formal requirement to seek planar constructions in preference to 
others. Yet it is almost invariably presented in modem accounts as the principal 
objective of problem solving throughout the ancient tradition. 10 Is there evidence 
that the ancients ever actually subscribed to this rule? Pappus* remark indicates 
at once that neither Archimedes nor Apollonius felt bound by it, for their own 
construct ions arc singled out for criticism. One can safely assume that the suitable 
alternative constructions could have t>cen produced, had either geometer admitted 
the need. 11 The remark relative to Archimedes is especially confusing, first in 
that it appears to misconstrue what a “solid** construction is, and second in that 
the alternative construction later given by Pappus employs conics, and thus is 
itself of the solid kind. 12 Surely Pappus meant to object to the appeal to the 
nettsis . on the grounds that when its replacement by a solid construction is 
possible, as here, that would be preferable. But then, this would not illustrate 
Pappus* chief |>oint about maintaining the appropriate boundaries between (lie 
planar, solid, and linear classes. Either Pappus is following an addled source, 
or else he is hetraying a certain confusion on his own part. Pappus* responsibility 
for this passage is suggested by the fact that it is an insertion into his general 
account of the classification of the problems, but only in one of its appearances 
(text // cited above), not the other (A). 

A review of the solutions devised for problems reveals little if any concern 
over this formal issue. Hippocrates constructs his third lunute by means of a 
nrusis , with not the slightest indication that a planar method (which can be given 
here) is to be supplied. 11 In the 4th century geometers attacked the problem of 
the cube duplication with a host of different techniques, including the intersection 
of solids, the construction of special curves, and the use of mechanical motions. 14 
As befits an early stage in the technical development of the field, the ambition 
is to find solutions by whatever means one can, to conceive of new methods 
attd explore their potential freely, but hardly to restrict artificially the domains 
within which one or another method is deemed valid. To be sure, Euclid's 
Elements investigates only one segment of the field of problems, those construc¬ 
tive by planar methods. Hut that neither exhausted the planar field, as we become 







346 Ancient Tradition of Geometric Problems 


Appraisal of the Analytic Field in Antiquity 347 


fully aware through the investigations of Apollonius, nor did it deter Euclid from 
pursuing extensive studies of solid problems and the theory of conics. 35 The 
successful discovery of a solid construction for the cube duplication did not 
discourage geometers after Mcnaechmus from the search for alternative solutions 
of the linear kind. When Nicomedcs castigates Bratosthencs over his method of 
cube duplication, it is not for reasons of formal correctness, but rather for practical 
feasibility.* These 3rd~ccntury geometers share an interest in finding new me¬ 
chanical procedures for solving geometric problems, and this seems to distinguish 
them from the geometers of the preceding century; but the formal ideal voiced 
by Pappus docs not concern them. 

The class of nettsis constructions undergoes a notable transition during the 
3rd century. In the work of Archimedes they arc used in the study of spirals, 
as auxiliary to the formal proofs of the properties of the tangents to these curves. 
Given that Archimedes is often sensitive to matters of formal procedure, as one 
sees in his delicate management of the introduction of his mechanical method 
in his Quadrature of the Parabola and the Method, this surely attests to his 
acceptance of nettses as formally proper. Blsewhcrc he solves the problems of 
(he inscription of the heptagon and the triscction of the angle via nettses, and 
geometers in the generation after him follow his example in devising nettses for 
the cube duplication and defining curves like the conchoids via nettsis condi¬ 
tions. 17 But with Apollonius a new element enters. Although he himself is 
credited with an alternative construction of the cube duplication via nettsis, he 
is also said to have used a method via conics for solving this problem. His On 
Nettses takes up only eases of the construction which may be effected by planar 
methods. Among his other analytic works, a division between planar and solid 
methods is indeed in evidence. 'Hie writings On Planar Loci and On Tan^encies 
deal only with planar constructions, while others, like the Section of an Area , 
the Section of a Ratio, and the Determinate hn i, seem to have their principal 
role within the study of problems on the construction of conics. *" It is thus clear 
that around Apollonius* lime (lie field of geometry had matured to (he effect that 
it had become important to regularize the formal methods of problem solving. 
The investigation of planar and solid problems as distinct classes would carry 
the expectation of discovering in detail the relative ranges of these methods, as 
would the effort to reduce other kinds of constructions, like the neuses, to one 
or the other class. 

In this way the relative ordering of the different kinds of problems served to 
guide the research efforts of Apollonius and his contemporaries. 39 Other methods, 
such as the generation of special curves via motions, would continue to be of 
interest, sincp problems like the circle quadrature and the arbitrary division of 
the angle remain outside the scope of both the planar and the solid methods. 
But even with this, it is not clear that Pappus* rule of method had attained formal 
recognition, for we may note that Pappus himself does not abide by it. For 
instance, having prefaced his section on methods of cube duplication with the 
statement about the classification of problems (A), he goes on to present four 
different solutions: two are mechanical (the methods of Eratosthenes and of 


Hero) and two arc linear (those of Nicomedes and of Pappus himself)* 40 Yet 
Pappus quite conspicuously informs iis that the problem is of the solid type. 4 
Apparently, neither he nor Hero, who also states the solid nature of this problem, 
senses that in presenting solutions other than solid ones he has committed a 
formal error. 

llicrc is a much dcc|>cr difficulty, however: in the context of the Greek 
geometry the formal rule proposed by Pappus and the classification scheme 
corresponding to it arc unworkable. If one can produce a planar construction for 
a problem, that docs indeed establish the planar character of the problem. But 
the discovery of a solid construction for a problem does not yet suffice for 
securing the solid classification of this problem; one must go on to demonstrate 
(hat no planar construction is possible. Although comparable impossibility proofs 
were an essential feature of the ancient theory of irrational lines, we have no 
grounds for supposing that the Greek geometers ever did or could set forth the 
kind of proofs required for the classification of problems. The forms ultimately 
worked out in the 19th century depend on algebraic techniques of a sort alien 
to the Greek geometric approach. 0 Despite this, Pappus and Hero seem perfectly 
comfortable in asserting the solid nature of the problems of cube duplication and 
angle triscction and the linear nature of the problem of the generalized angle 
division, oblivious, it would appear, even to the realization that such a claim 
required a proof. Indeed, Pappus prefaces Book HI of the Collection with scornful 
remarks on the attempt by a colleague to solve the cube duplication via planar 
methods. 41 To be sure, the method is unsuccessful, being at best a procedure 
for the arbitrarily close approximation of the solution via a recursive planar 
construction. But Pappus speaks almost as if the very notion of trying to find 
such a solution for the *‘solid** problem is absurd per se. Like the mathematical 
authorities of a much later age who effectively ruled against the possibility of 
the circle quadrature a century Ircforc the transcendentality of the problem was 
actually demonstrated, 44 he appears to have accepted the long record of failed 
attempts as the basis for affirming the solid character of the cube duplication: 

Admitting the problem to be solid, they [e.g., Eratosthenes, Philo, and Hero] 
effected its coastmction only by means of instruments—in agreement with Apol¬ 
lonius of Perga who also effected its analysis by means of the sections of the cone, 
while others |did so| by means of the solid loci of Arislacus, but no one by means 
of properly so-called planar |incthods|.*’ 

However convenient the classification might be, we would surely have appre¬ 
ciated from him some recognition that the formal placement of a solid problem 
like this is an open question subject to proof. 

The situation with respect to loci is subtly different. A M planar locus” is by 
definition a straight line or a circle; a **solid locus” is one of the conic sections. 
If, then, the determination of a problem of locus results in one of the former, 
the hrcus is planar; if one of the latter, it is solid. There can be no ambiguity 
here, so that the classification of loci and problems of locus into planar, solid, 
and linear categories is perfectly feasible. As indicated above, the terminology 




348 Ancient Tradition of Geometric Problems 


Appraisal of the Analytic Field in Antiquity 349 


is already well attested with Apollonius and well suits the aim of systematizing 
the study of constructions in the analytic works by him and his contemporaries. 
What remains difficult to explain, however, is the rationale of the terms “planar” 
and “solid** in the instance of loci, unless we arc to accept the commentators' 
very weak proposals, as considered above. It seems to me that an account of 
the following sort is possible. Well before Apollonius, one already knew that 
loci of the former sort, namely lines and circles, had their special domain of 
application toward the solution of problems in which a line segment must be 
found to satisfy a specified second-order relation with respect to given lines. 
Similarly, loci which are conics were known to be applicable toward the solution 
of problems in which the line segment must satisfy a third-order relation. This 
aspect of the applicability of these two types of loci could justify naming them 
"planar” and “solid,” respectively, even if one did not have a clear proof that 
all problems expressible via such second- and third-order relations can be solved 
via loci of the former and latter types, respectively. The naming would surely 
be descriptive, rather than normative, and would be convenient for focusing 
research into (he relative domains of problems solvable via the different methods 
of construction. 

It is not difficult to see how one might go on to extrapolate this viable scheme 
for problems of locus to the whole field of geometric problems, in this way 
obtaining (he classification used by Hero and Pappus. Although this move brings 
with it the logical difficulties just mentioned, (lie commentators seem unaware 
of this fact. It is thus possible lhat one of them bears responsibility for introducing 
die defective conception of the classification scheme, rather than its having been 
a fixture within the earlier research tradition. Since neither Proclus nor Eulociits 
follows this scheme, it would appear not to have received a conspicuous statement 
in the works of the earlier geometers or in the later compilations by Geminus 
and Mcnclaus. Far from being the "standard” ancient view, then. Pappus' 
tripartite division of problems and the criterion for the choice of construction 
techniques which accompanies it emerge as a minority opinion held by a few 
of the late commentators through their misconception of the nature of the 3rd- 
century enterprise of problem solving. 

PROBLEMS, THEOREMS AND THE METHOD OF 
ANALYSIS 

The distinction between problems and theorems is maintained meticulously by 
the authors of the principal treatises of classical geometry, Euclid, Archimedes, 
and Apollonius, as well as in the miscellaneous materials preserved by the lalcr 
commentators, Hero, Pappus, and Eutocius. As a matter of form , a problem is 
cast as an infinitive expression seeking the construction of a geometric term in 
a specified .relation to other given terms (e.g., "to construct the triangle whose 
sides arc three given line segments”). 4 * By contrast, a theorem is typically set 
in the form of a conditional asserting a'prhperty of a specified geometric con¬ 
figuration (e.g., "if in any triangle two lines arc drawn front the endpoints of 


its base and meet within it, then the lines will have a combined length less than 
that of the two remaining sides”). 47 A point not missed by the ancient com¬ 
mentators is that the theorem refers to a general class of entities (e.g.. "any 
triangle”), whereas the problem usually results in the production of a unique 
figure. Indeed, it may happen that the problem cannot be solved unless certain 
auxiliary conditions arc met. One thus typically supplies a “diorism” asserting 
those conditions (in the above ease, "where the three lines arc such that no two 
have a combined length less than the third”). This might he motivated through 
a separate Ihcotctn (as in Euclid, “in any triangle the combined length of any 
two of its sides is greater than its third side”) 4 * or alternatively, usually in the 
context of an analysis of the problem, through an allusion to a condition for a 
certain step of the construction (e.g., that two given circles must have a point 
of intersection not lying on the line joining their centers). When the problem 
has many solutions, it may be expressed as a theorem of locus (e.g., “the points 
whose distances from two given points have a given ratio lie on a given circle”), 
or as a locus problem (e.g., “to find the locus of points such that.. . ”). 4 * Such 
loci provide an important instrument for the solution of problems. For instance, 
the problem of constructing the triangle of three given sides might alternatively 
he found via the intersection of a circle (the locus of points whose distances 
from two given points are in a given ratio) and an ellipse (the locus of points 
whose distances from the same two given points have a given sum). Of course, 
since tills particular problem permits of solution within the context of elementary 
geometry, one could not here wish to appeal to the more advanced methods of 
conics, ns this alternative construction does. 50 

Ill esc illustrations indicate that jrom the p urely. JonnaLviewpointJhc dis- 
tinction between problems and theoremshi largely artificial. One can easily recast 
aiiy problem as a theorem, merely by incorporating into the protasis of the 
theorem all the details of the construction of the problem. Conversely, a theorem 
might l>c rephrased as the demand to produce a construction or to discover some 
property of a given figure. In most instances, there would appear to be a natural 
preference for one form over the other. In the case of propositions of locus, 
however, the distinction is not entirely clear. Like problems, they require a 
construction; but like theorems, they aim to prove a property of what is con¬ 
structed. namely, its identification as a given line, circle, conic, or other figure. 
Pappus and Proclus thus reserve a third category, “porisms.” in separation from 
those of problems and theorems, 41 and I suspect that this is nothing other than 
the class of loci. The commentators arc neither clear nor convincing in explaining 
how porisms differ from problems, hut they do assure us that Euclid's Porisms 
was a compilation of theorems of locus. 1 would thus suppose the "porism” 
was Euclid's lenn for what later geometers designated as “locus,” and that, not 
perceiving this, the later commentators attempted their own explanation oil etym¬ 
ological grounds. 

A formal treatise in geometry might thus he organized as a body of theorems 
presented in the order of their deductive dependence. As Proclus observes, Euclid 
strikes a roughly even halancc between theorems and problems in his first book. 47 



350 Ancient Tradition of Geometric Problems 


Appraisal of the Analytic Field in Antiquity 351 


In this way, the problems often serve as justification for the introduction of 
auxiliary terms in later propositions. For instance, the theorem that the angles 
of any triangle equal two right angles (I, 32) requires the introduction of a line 
parallel to a side of a given triangle and passing through one of its vertices; this 
can be effected through the construction set out in the preceding problem (I, 
31). Commentators, both ancient and modem, have thus found it natural to 
assign to the Euclidean problems the special role of securing the proofs of 
existence of the geometrical entities investigated in the theorems/ 3 Wc shall 
consider this issue further below. But one should note that this view offers, at 
j best, an incomplete picture of the diverse role of problems within (he ancient 
* geometry. Some problems arc included not for their applications in later theo¬ 
rems, but for their intrinsic interest. Book IV and most of Book XIII, for instance, 
are devoted to the exposition of problems: the inscription of regular polygons 
in the former, that of the regular polyhedra in the latter. Here, the subordination 
is reversed, in that any theorems enter as auxiliary lemmas toward the construc¬ 
tions. 54 Pappus* treatment of the angle trisection suggests a provenance in a 
deductively ordered structure of solved problems: the triscction assumes a neusis 
whose solution as a solid problem is known from a prior lemma; the latter, in 
turn, assumes the construction of a hyperbola of given asymptotes, solved in 
another lemma; this refers back to the construction of the conic sections answering 
to given parameters, as solved by Apollonius in Book I of the Conics / 5 Clearly, 
the structure of these problems is determined by the deductive requirements of 
the problems themselves, not by the demands arising from their use in other 
theorems. Indeed, Apollonius remarks that theorems like those expounded in 
Conics III have their principal value within proofs dealing with locus problems. 56 
It is noteworthy that problems are few in number in the first four books of the 
Conics . As wc have seen, the appearance of the hyperbola construction in II, 4 
is due to an interpolation by or after Eutocius; (he sections of problems at the 
end of Book l and closing Book II cannot easily be viewed as contributing to 
the deductive needs of the theorems in other pans of the Conics . Their removal 
would in no way affect the logical structure or soundness of the main body of 
theorems. 57 The same may be observed of Euclid’s Book 1. The validity of its 
theorems does not depend on the presence of the problems. ,R Were one to 
segregate the problems, however, the proofs of their constructions would require 
appeal to the theorems in the rest of the book. One may see that the three 
postulates of construction—to draw the line through two given points; to extend 
a given line indefinitely; and to draw the circle of given center and radius—are 
stated in the form of problems whose construction may be assumed as irreducible, 
and so may initiate the ordered sequence of problems. By contrast, the two 
remaining postulates—the equality of all right angles and the nonparallelism of 
two lines when the interior angles formed with them by a transversal line suvn 
to less than two right angles—are stated in the form of theorems and arc intro¬ 
duced as premisses in theorems in Book I. 59 

Further examples of this kind could be supplied in profusion. They reveal 
that orobfem solving was the essential nart of the geometric enterprise marked 


off by the works of Euclid, Apollonius, and those in their tradition, and that for 
them the compilation of bodies of theorems was an effort ancillary to this activity. 
It is important to realize this as one turns to the views proposed by ancient 
thinkers on the nature of problems and theorems and their relative significance 
for geometry. Pappus and Proclus draw from a centuries-long tradition of dis¬ 
cussion of the definitions of these and associated terms, of which the following, 
from Pappus, is a representative sample: 

Those wishing to discriminate more precisely the things sought in geometry... think 
fit to call a “problem” that in whose case it is proposed {prohalhin) to make or 
construct something, but a “theorem 0 that in which, on the supposition of certain 
things, one investigates ( theorem ) what is implied by them and is contingent on 
them in every way. Among the ancients, some call |ihcni| all problems, others 
|call them all| theorems/ 1 

On this latter division of opinion, Pappus tells us no more; but Proclus reports 
the positions of half a dozen writers widely distributed over the span of antiquity. 
As one might anticipate, those emphasizing the role of theorems draw their 
inspiration from the ideas of Plato: since the objects of geometry are eternal, 
unchanging verities, it is improper to speak of one’s bringing such things into 
being, constructing them, and so on, the way one typically does in a geometric 
problem. 61 Rather, their study is the contemplation of what they truly are, and 
thus only the form of the theorem is appropriate to this discipline. Proclus ascribes 
this view to Plato’s nephew Spcusippus, as well as to a mathematician named 
Amphinomus. 62 Much the same view is assigned to Plato himself by Plutarch, 
on the authority of the Platonic essay by Eratosthenes 63 ; even if the incident of 
Plato’s chiding Eudoxus for his approach to solving the cube duplication is surely 
apocryphal, this conception is indeed faithful to Plato’s own thought on the 
nature of geometry. As director of the later Academy and exponent of Neopla¬ 
tonism, Proclus naturally finds his sympathies allied with the same view. In 
agreement with Geminus, he makes it the aim of geometry to move through 
problems to theorems, as it were, rising from sensibles to ideals. 64 

By contrast, writers closer to the activity of geometers emphasize the contrary 
view. According to Proclus, the ’’mathematicians in the following of Menaecb- 
mus” classed all the propositions as ’’problems” (prohlemata ) 65 : 

while the problem (jjrohole ) is of two sorts. Sometimes the thing sought is produced 
(porizein); but sometimes by taking what has been delimited we sec either what 
it is, or what kind of thing it is, or what property it has, nr what relations it has 
to something else. 

Here we perceive the same distinction which results in the separation of locus 
problems (and “purisms”) from the body of problems of construction, although 
Menacchmus hardly subscribes to Euclid’s terms. Nevertheless, it is an important 
insight that the methods of finding constructions might be applied toward the 
discovery of properties of given configurations. Proclus does not provide the 
rationale behind Menaechmus* view, although it would appear to be epistemo¬ 
logical. in contrast with the ontological nosit inn of the Platonists. That is. the 



352 Ancient Tradition of Geometric Problems 


Appraisal of the Analytic Field in Antiquity 353 


"problem” is the superior vehicle in the heuristic effort of gaining knowledge 
about geometric figures. Proclus attempts a compromise by admitting that we 
must engage in an activity which employs a vocabulary of construction, but that 
geometry deals only with the "intelligible matter” of figures, rattier than with 
their “sensible matter." as one does in mechanics."'The actual being of abstract 
geometrical entities, however, is subject neither to generation nor to change of 
any sort. Proclus, through his preoccupation with the Platonic doctrines on the 
nature of being, seems to have missed the key issue of heuristic method. 

Another mathematical writer. Carpus of Antioch "the mechanician,” defends 
the priority of problems on pedagogical grounds. Else where he is known to us 
for an account of the mechanical work of Archimedes and for having solved the 
circle quadrature via "the curve called simply *of double motion*.” M To Carpus, 
problems have an obvious advantage over theorems in their being simple to state. 
Proclus cites, at some length, certain views drawn from his work ”on the subject 
matter of astronomy”: 

on the distinction of these, lie says that the problematic genus is prior to lhai of 
the theorems. For the subjects about which properties arc sought arc foiiud through 
the problems. Moreover, the statement {proum: i) of the problem is simple and 

without further need «>r any specialized understanding_Hut that of ilic lliroicm 

is effortful and demanding of great precision and cx|K’r 1 discrimination, in onlcr 
that it appear neither more nor less than (lie innh.“* 

Carpus illustrates this point in the particular context of the problems and theorems 
which open Euclid’s Book I, and so has in mind the pragmatic question of how 
best to present geometric materials to learners. Proclus concedes the greater 
effectiveness of the problematic form in this case, as also in the case of the 
application of geometry to the mechanical arts. But he musters support from 
Gcminus for the view of the greater perfection of theorems, on the grounds, as 
noted above, that these better capture the essence of abstract geometric truth. 
Proclus thus seems not to sense the significance of another statement by Carpus, 
that the usual methods for problems are well suited for “the more obscure cases. ” 
From the viewpoint of geometric research, this is surely the striking advantage 
of the problematic form: that one may, with relative ease, articulate a problem 
whose solution is not known and in this way define a direction for gaining new 
knowledge in geometry. By contrast, the statement of a theorem must already 
incorporate the full description of what has been discovered. The theorem, then, 
is fhc appropriate mode when one is concerned with the formal organization of 
known results, rather than with the discovery of what is not yet known. In their 
respective positions on the evaluation of the problematic and theoretic forms, 
Proclus and Carpus thus signal the contrasting objectives appropriate to their 
differing manners of involvement *in geometric studies. 

Proclus calls attention to another distinction between problems and theorems*": 

According to //modulus’", the theorem is distinguished from the problem in that 
the theorem seeks what is the properly predicated of iis subject matter, while the 
nmhlrm l<rrktl what is to nisi fin nntrr Hull snmrthinp hr Whrnrr also ar. 


cording lo Posidonius/ 1 ihc one is defined as a proposition according to which one 
seeks whether it is or is not |possible], the other’ 1 a proposition in which one seeks 
what it is or what sort of thing it is ... For it is different either to seek simply 
and indefinitely whether it is (possible to construct a liiic| at right angles from this 
pnini to this line, 01 to investigate \thcotctn\ what is I the nature of| the line at 
light angles. 

Despite the highly elliptical terms, it is dear that both views assign to probtems 
an existential function. In the former case, they effect the configuration such 
that a specified condition may be said to obtain. In the latter, they determine 
whether or not a construction satisfying the specified condition is possible. They 
thus accord quite comfortably with an account in the manner of modem logic, 
which would transcribe theorems via universal quantifiers, but problems via 
cxistcniial quantifiers. Indeed. Proclus subscribes to this view of problems in 
his own reconstruction of Euclid's dialectical purposes in first setting out three 
problems of construction (I, I -3) before taking up the first theorem on congruence 
of triangles (I, 4): 

Without a prior construction ol the triangles and a prodticlion of their genesis, how 
could he presume lo tench altouf their csscnlial properties and of the equality of 
their angles and their sides? . . For let someone, happening along before the 
making of those things, say that if two triangles have n property like that, they 
shall also have this one in every case. Then, would it not be easy for everyone to 
respond to him: For do we at all know whether a triangle is capable of being 
constructed? . To cul off these difficulties in advance, then, die author of the 
Mntwtux has also handed down (he construction of the triangles.. .and lo these 
he quite reasonably attaches die theorem by which is proved |thc property of 
congruent triangles asserted hcir| " 

I lowcvcr nicely this account suits the needs raised by Proclus* Platonist ontology, 
il hardly presents a convincing portrait of Euclid’s intent. If he had indeed wished 
to affirm the existence of triangles, he need only have taken three arbitrary points 
and then introduced the three tine segments which connect them two by two in 
accordance with his first postulate. The assumption of arbitrary points or lines 
as "given” is not founded on any explicit postulate, but is in fact an clement 
in practically all of Euclid’s problems. For instance, in I, I, the line segment 
which is to serve as the base of the couslmcled equilateral triangle is such a 
given term. What is especially inappropriate about this construction as an alleged 
proof of the existence of triangles is that it brings about only one particular kind 
of triangle and might thus risk misleading the learner into conceiving of this 
special case, when in the context of general theorems on triangles, such as I, 
4, he ought to conceive a case better representative ol the general class. But as 
a simple yet interesting first exercise in the mobilization of postulates and other 
general principles for working out a formal geometric proposition, the problem 
Euclid has chosen to open Book I is beyond reproach. 74 

Il is of course an important insight on the part of Proclus and the commentators 
on whom he depends to perceive this nuance in the logical conception of prob- 

Irtitc nnd u/r* would rrrfninlv mil urich In flrnv that Fur I id and nltl£r aPOfflfters 




354 Ancient Tradition of Geometric Problems 

might themselves have been aware of it. But as we have seen in the many 
examples referred to earlier, this view cannot cover the full range of contexts 
served by the introduction of problems in the ancient geometry. Indeed, this 
existential aspect is strongly marked only in the case of “diorisms/' which 
articulate the limits on the solvability of problems. Note, however, that the 
condition of solvability for the problem of constructing a triangle of given sides 
(1, 22) is framed by Euclid as a theorem (I, 20): that any two sides of a triangle 
have a combined length greater than the third side; not as a problem. 1 * Euclid 
does not actually show how this result secures the existence of the point of 
intersection of the two circles by which the construction of the triangle is effected 
in the subsequent problem. It has thus been left to Proclus and other commentators 
to fill this gap. 7 * In view of Euclid's omission, one becomes doubly wary of the 
suppositions that he appreciated the need for existence proofs in cases like this, 
and that he inserted the problems to serve this need.” 

Throughout the present study, we have seen the heuristic power which the 
method of analysis afforded the ancients in their search for solutions to geometric 
problems. The most detailed account of this method surviving from antiquity is 
given by Pappus as an introduction to his survey of the analytic corpus in 
Collection VII. To this wc may add two very brief statements on the nature of 
this method, the one in a scholium prefacing some alternative proofs to the 
opening theorems of Euclid's Book XIU, the other from Hero's commentary to 
Euclid's Book II as extant in the Arabic translation by al-Nairlzi™ A comparison 
of the three versions reveals some important aspects of Pappus’ account which 
seem not to have been considered in the profuse discussion of it by modem 
scholars. 79 

Pappus opens with a statement summarizing the nature and objectives of the 
analytic corpus (topos analyomenos):™ 

The so-called "analytic |corpus|‘ ... is a certain special body prepared next after 
the making of the common elements for those wishing to acquire a power in lines 
li e., lines and curvcs| conducive to the finding of problems proposed to them, 
and it has been compiled as useful for this purpose alone. It has been written by 
three men: Euclid, author of the Elements, and Apollonius of Perga, and Arislaeus 
the Elder, and it adopts the method according to analysis and synthesis. 

He now sets out to characterize this special method for problem-solving: 

{Pappus) Now analysis is a way from what is sought, as if admitted, through the 
things that follow in order, up to something admitted in the synthesis/ 1 

{Scholium) Now analysis is n taking of what is sought, as if admitted, through the 
things (hat follow, up to something admitted as true." 

{Hero) Now analysis is, when any question has been proposed, that we first set it 
in the order of the thing sought, which has been found | !|, and we will then reduce 
(it to something) whose proof has already preceded. Then therefore it is manifest, 
we say. that the thing sought has now been found according to analysis/ 4 

The statements by Pappus and the scholiast are obviously in virtual literal agrec- 


Appraisal of the Analytic Field in Antiquity 355 

incut. An actual dependence of the scholiast on Pappus is entirely possible; but 
the confomiity in sense (if not in specific wording) with Hero reveals that at 
least this much of the account has a firm precedent in sources older than Pappus. 
Prom here. Pappus amplifies thus: 

{Pappus) For in the analysis, having hypothesized what is sought as if already in 
effect, we examine that from which this results and in turn the antecedent of that, 
until proceeding in this backward manner we come down opposite some one of 
the things already known or having the order of first-principle. And such a method 
we call * analysis/ as being like a backward resolution. 

Through our experience of the ancient uses of analysis, we should have expected 
it to be described as a searching through the geometrical consequences deduced 
from what is sought, on the hypothesis that the latter is already in effect. This 
is the natural reading of the opening lines, even if the term “things that follow” 
{to akobutha) need not bear the stronger sense of logical deduction in a context 
like this one/ 4 Pappus, however, seems to view the procedure in quite a different 
manner, as a finding of successive antecedents, so (hat the result initially hy¬ 
pothesized ends up being a deduction from the derived sequence. But before 
pursuing this further, let us note that Pappus’ remarks have the appearance of 
being a gloss on a statement on the analytic method received from a prior source. 
A similar pattern emerges in the next section of his account. The two alternative 
accounts provide the anticipated analogues to their statements of analysis: 

{Scholium) But synthesis is a taking of the thing admitted through the things that 
follow up to something admitted as true.” 

{Hero) Hut synthesis is, that wc begin from a thing known, then wc will compose, 
until the thing sought is found. Ilms, the thing sought will then be manifest 
according to synthesis. 

Pappus omits any such corres|H)ndmg line, but instead provides llie complement 
to his own amplified statement on analysis: 

iPappus) Hut in the synthesis, in reverse order, having posited as already in effect 
what had been obtained last in the analysis, and having ordered in the natural 
manner as consequents what there |wcre ordered as| antecedents,"* and having 
composed them to each other, wc at last arrive at the construction of what is sought. 
Amt this we call ‘synthesis/ 

For Pappus, then, the difference between the two methods is merely one of the 
order of exposition. Despite his earlier insistence on the singular significance of 
analysis for the study of problems, he now accords with Hero and the scholiast 
in considering its application in the instance of theorems. To establish a theorem 
analytically, then, Pappus would hypothesize as true the desired conclusion A 
and then reason, “A will follow, if B holds; and B, if C; etc." Arriving at 
something known to be true (e.g., a previous theorem or a postulate), he can 
then affirm the truth of A. Now, Hero and the scholiast go on actually to work 
out demonstrations in the double mode for a series of theorems/ 7 In the former 
instance, this results in alternative proofs for eight of the first 10 theorems in 







< .. wa • luaiavaMtl 

Euclid's Book II; in the latter, for the five theorems which open his Book XIII. 
The theorems are all quite straightforward, and both writers treat them in the 
same manner. Ear from adopting the mode suggested by Pappus, both frame the 
analysis as a sequence of deductions leading from the hypothesis of the result 
to be proved and terminating in something known to be true. In effect, their 
analyses arc nothing other than regular proofs of the converse of the proposed 
theorems. Furthermore, one may note that both writers share an interest in 
securing their theorems without recourse to the geometric diagrams, this being 
possible here in that only the formal manipulation of identities is at issue. Such 
affinities between these two treatments must surely indicate mutual dependence, 
cither through common authorship or through the scholiast's referring to Hero 
for his model. In the light of our earlier observation of the agreement between 
the scholium and the opening lines of Pappus' account, this would suggest that 
it is Pappus who has borrowed from a source (for instance. Hero himself or a 
writer following him) to establish a familiar context for the elucidation of this 
method, ITic remarks which he attaches to it, however, would reflect his own 
interpretation of the ancient geometry, but have no particular standing within 
the older tradition of mathematical commentary. 

A fundamental difficulty in Pappus* view of the order of deductive sequence 
in die analysis is that, if lie were correct, there ought to be no need for the 
synthesis; for the analysis of itself would constitute a demonstration of the 
theorem. It happens that a treatment adopting precisely the form implied by 
Pappus appears among his lemmas to Apollonius* Determinate Section.** In the 
context of a s|>ccificd figure, it is to be proved that a stated relation A obtains; 
the writer passes through a scries of steps of the form “B is the same as C” 
and "if D obtains, it will Ik necessary alternately to seek whether E obtains.’* 
He ultimately hits upon a relation known to obtain. In his paraphrase of this 
lemma, Heath speaks of it as "a case of theoretical analysis followed by syn¬ 
thesis,” and after the analysis he adds the words, "The synthesis is obvious." 
To be sure, this lag line appears now and again among the analyses in the 
Collection.** But in the present case it is purely Heath’s addition; for it is not 
used here by Pappus, and there is nothing which overtly indicates that this proof 
is intended as an analysis or that a synthesis ought to Ik supplied. Thus, if the 
writer of the lemma is following the procedure prescribed by Pappus, replacing 
each derived relation with another equivalent to it, he treats the synthesis not 
merely as "obvious," but as superfluous. 4 " 1 

One can see how Pappus might construe a proof technique of this sort to be 
a form of analysis. Reflecting on passages in the philosophical literature where 
the analytic manner is pictured as a search for antecedent premises or conditions, 
he might then adopt this as his own view, as he sets out to amplify the sketchy 
remarks made by earlier mathematical commentators. The following passage 
from Aristotle’s discussion of the nature of moral deliberation (houleusis) is 
especially interesting in this connection 91 : 

The statesman d<ics not deliberate over good order, nor does anyone else over the 

end |to be achieved). Bui positing the end , ihcy investigate the questions of how 


and through what things it shall be. If it appears 10 result from several things, they 
investigate through what it will result in the easiest and best manner; but if it seems 
to result from a single thing, they consider how it shall result from that, and what 
that in turn tcsidts from, until Ihcy arrive at the first cause which is last in the 
older of discovery. For the dclibcrator seems to seek and to analyze in the said 
manner, just as in the case of a diagram |i.c., geometric figure or proposition! 

... and that what is last in the analysis is first in the genesis. Now if Ihcy happen 
upon something impossible , they leave off; for instance, if it requires money, but 
this cannot Ik* provided. But if it appears possible, they try to do it. 

Forms of the tcims indicated in italics appear in Pappus* account of analysis as 
well. 9 ' It fixes the method as a search for antecedent causes in a way that examples 
drawn from the mathematical literature never would. Even Aristotle's consid¬ 
eration of (he two alternative outcomes of the analysis figures prominently in 
Pappus* subsequent remarks. Wc arc thus readily led to suppose that Pappus 
drew the basis of his account of analysis from the philosophical literature. Of 
course. Pappus does not cite Aristotle here. 9 ' But nowhere in the Collection does 
he cite any philosopher other than Plato, and that only once for a philosophical 
view.* 4 The several references to the "five Platonic solids," that is, the regular 
polylicdra tin Pappus’ Book V), can hardly be viewed as citations of Plato. One 
can discern several Aristotelian doctrines in Pappus* Commentary on huclid's 
Hook X , even though he diKs mil expressly cite Aristotle there cither. 4,1 It seems 
to me most probable that Pappus could pick up such general views through the 
medium of commentators like Geminus and others, conversant with a syncre- 
tizing form of Platonism. 9 " In such a case, Pappus himself might not be fully 
aware of the ultimate provenance of some of his views. 

One thus approaches the concluding section of Pappus' account with suspi¬ 
cions that it, loo, despite his access to the whole ancient corpus of analysis, 
may present not a distillation of that ancient tradition, but rather a rephrasing 
of standard philosophical views. He is concerned here with distinguishing be¬ 
tween llie analysis of theorems and that of problems: 97 

The class of analysis is of two sorts: the one which is called ‘theoretic* seeks what 
is tme. while the other which is called ‘problematic’ is productive (poristikon) of 
wliut has been proposed. In the case of the theoretic kind, having hypothesized 
what is sought as being and as tme, ami then having gone forward through the 
things which follow in order, as true and as being in accordance with hypothesis, 
up to something admitted, then if that thing admitted is true, so also will the thing 
sought be true, and the proof will be converse to the analysis; hut if wc come upon 
a thing admitted as false, so also will the thing sought be false. 

In the case of the problematic kind, having hypothesized the thing proposed as 
known, and then having gone forward through the things which follow in order 
as true up to something admitted, then if that admitted thing is possible and 
producible, what those in mathematics call ‘given’ ( dothrn ). so also will the thing 
proposed be possible, and in turn the proof will be converse to the analysis; but 
if wc happen upon something admitted as impossible, so also will the problem be 
impossible.*" 





358 Ancient Tradition of Geometric Problems 


I 


l 



H 


And a ‘diorism* is a preliminary distinction of when and how and in how many 
ways the problem also will be possible. 

So much then concerning analysis and synthesis. 

This amounts to hardly more than an elaboration of the preceding remarks in 
the obvious way. Pappus remains uncertain about the order of implication in the 
analyses, and in the strict sense he would require all steps to be simply con¬ 
vertible. Otherwise, he could not maintain both that the truth of what is derived 
implies the truth of (he hypothesis and also (hat its falsity implies the falsity of 
the hypothesis. As we have seen, the effort of logically converting the steps in 
the analysis of a problem to obtain its synthesis is the basis for discovering the 
conditions under which the solution is actually possible* and so leads to the 
articulation of the diorism. But what Pappus has to say about this critical element 
of the method is so extraneous to his account that the modem editors typically 
bracket this line as a later interpolation." Similarly, the extremely subtle role 
of the “givens" in the analysis, as we have observed in Chapter 4, is barely 
hinted at in Pappus’ feeble aside. Applying insights derived from the modem 
logical investigation of heuristics, Hintikka and Remes have perceived nuances 
in the logical status of auxiliary constructions in the analysis of figures. 100 From 
Pappus' account one may gather, however, that the commentator was at best 
but subliminally aware of such difficulties. 

What Pappus emphasizes is the distinction between the theoretic and prob¬ 
lematic forms of analysis. But examined more closely, one begins to see that 
his notion of an analysis of theorems is gratuitous. To the extent that he is 
viewing the expository form of the method, rather than its heuristic role, he 
seems not to sense that such analyses are entirely redundant 101 ; for consider any 
one of the cases worked out by Hero or the scholiast to Elements XIII, for 
instance, (he analysis and synthesis of the theorem XIII, I, to the effect that if 
a line is divided into segments jc and y in extreme and mean ratio (i.c., y : x = 
X : x + y), then the square of '/* ( x -» y) T x is five times the square of Vi (.v 
+ y). In the analysis one adopts the hypothesis that (2 l x? = 5 z 7 and then 
deduces through the condition that x and y are in extreme and mean ratio that z 
= l A(x + y); in the synthesis one starts from z = Vi ix T y) and deduces that 
(z + xf = 5z 2 . Clearly, the presentation of the analysis here is completely 
artificial. In the case of the more complicated geometric configuration given by 
Pappus in a lemma toward the construction of a case of Apollonius’ three-circle 
problem, the steps leading to the desired proof might be less obvious, so that 
the artificiality of the analysis is less evident. 102 But what the commentators lose 
sight of is that such theorems arise within the context of other theorems or 
. problems which not only reveal what the claim made by the theorem ought to 
i be—that is not merely a matter of conjecture—but also will display the elements 
' of the desired proof. It is no wonder, then, that analyses of theorems are so 
rarely found in the mathematical literature. These are either artificial and un- 
I necessary (as in those given by Hero) or serve for finding proofs to theorems 
whose stated result is somehow known in isolation from the context which gives 


Appraisal of the Analytic Field in Antiquity 359 

rise to them. Note, furthermore, that in Pappus’ alternative outcome of the 
analysis of the theorem, where the derived result turns out to be false, the formal 
demonstration may be produced merely by negating the hypothesis to obtain a 
new assertion and then submitting the steps of the analysis, as already worked 
out. in proof of this. The term "reduction to the impossible" designating this 
form of proof was already familiar at the time of Aristotle, and doubtless even 
before then."” By its very name, it invites comparison to the method of “re¬ 
duction" used, for instance, by Hippocrates in the examination of the cube 
duplication, and discernible as a precursor of the method of analysis. The con¬ 
nection was not missed by the later commentators. 1 *** What this reveals is that 
for a large body of theorems, the analysis of itself yields a proof, so that no 
synthesis will be called for. Once again, the artificiality of Pappus' theoretic 
type of analysis and synthesis is apparent. 

It is not difficult to explain how Pappus came to advance this notion of theoretic 
analysis."” If we again consider the theorem of the scholiast, the result it es¬ 
tablishes is needed in XIII, 11 for showing that if a regular pentagon is inscribed 
in a circle of rational diameter, its side will be a "lesser" irrational line. 106 The 
construction here gives rise to the division of a given line into extreme and mean 
ratio and the consideration of the ratio of (r + x) 1 to i*, for r = Vi (x + y). 
One might well adopt the format of an analysis to determine what that ratio is, 
namely 5 : 1. But in doing so, one would be analyzing not a theorem, but a 
problem.' 01 We recall that Mcnaechmus had indicated two types of problems: 
those that lead to the construction of a figure having cerlain specified properties, 
and those that lead to the determination of certain properties obtaining for a 
specified figure. In the latter case, the result would naturally be stated as a 
theorem, of which the proof would be identical to the synthesis of what was 
earlier proposed as a problem. But having made this determination, one would 
have neither expository nor heuristic reasons for fabricating an analysis in ad¬ 
dition to the synthesis of this derived theorem. 

in this regard, special interest attaches to an observation made by Carpus in 
his advocacy of the su|>criorily of problems over theorems 10 *: 

And in the case of problems there is a certain single way. that found in common 
through the analysis, according to which by going forward we can achieve success. 

For in this way one hunts after the more obscure of the problems. But in the case 
of theorems the handling is hard to get hold of, as up to our limes, he says, no 
one has been able to offer a common method for the discovery of these, so that 
by virtue also of their facility the class of problems would be simpler. 

Although Carpus does not insist that analysis is a method appropriate to problems 
to the exclusion of theorems, that is the natural inference to be drawn from his 
remark. He is known to antedate Pappus, who cites him, and we would surely 
set him before Hero as well. For the effort by Hero and Pappus to accommodate 
the method or analysis to theorems, indeed to suggest in subtle ways that its use 
for theorems might even be prior to that for problems, would surely have pressed 
Carpus not to deny the existence of a proposal for a general method of inves- 




tl v j.&v rcwa^v^ ' J i *.t »r 


360 Ancieni Tradition ol Geometric Problems 


Appraisal ol llic Analytic l icld m Antiquity 361 


i; 

t: 


ligating theorems, but rather to question the alleged utility of analysis in these 
cases. Viewed thus in the light or Carpus* position. Pappus' account of analysis 
gains a dialectical dimension. By insisting on the relevance of this method to 
the study of theorems, he effectively neutralizes one of the more telling arguments 
on the side of the problems in Ihe continuing debate over the relative importance 
of problems and theorems. 10 * But he does this without convincing support from 
the mathematical literature, where the instances of theoretic analysis, rare on 
any account, all seem to be due to later commentators, like Pappus himself, as 
they supply minor lemmas to the great treatises of the older geometric tradition. 

Blinded by the glare of Plato's ontology, the later commentators seek to 
impose on the classical geometry an interpretation it does not easily sustain. 
Since problems seek to construct geometrical entities, that is, to bring into being 
things which are forever existing, or seek to cut, augment, diminish, or otherwise 
modify things which are in reality immutable, they must assume a position 
subordinate to theorems. For only the latter are philosophically correct in form, 
proving what is true of the ideal entities of geometry. Problems are allowed a 
role at the tentative stage when knowledge is incomplete, or for the convenience 
of learners. Or they might render the service of effecting proofs of the existence 
of terms needed in the formal demonstrations of theorems. Even the method of 
analysis, the method par excellence of problem solving, is transferred to theorems 
through the artificial ploy of “theoretic analysis," while in their quest for formal 
proofs Ihe commentators regularly choose to delete the analyses from older 
treatments cast in accordance with the double method."" In all these ways the 
later tradition betrays its insensitivity to the outlook and objectives of the older 
enterprise of geometry. To be sure, there are rare occasions wlicn the commen¬ 
tators somehow grasp the discrepancy. In his discussion of Euclid's Porisnis, 
for instance, Pappus observes 1 ": 


i 

I 

i 

i 

i 


...the ancients best knew the difference between these three kinds, as is clear 
from the definitions; for they said that a “theorem*' is what is put forward for the 
proof of the thing proposed, a “problem" is what is projected for its construction, 
and a "porism" is what is put forward for its production {porismos ). But this 
definition has been changed by the more recent |writcrs| who can't produce t/mr- 
iiein) everything, but use these elements and prove only this, that what is sought 
exists."’ but don't produce this. 


; This statement must discourage any supposition that the ancients conceived of 
problems merely as existence lemmas for Ihe proofs of theorems. In so distancing 
himself from those contemporaries who supposed this very thing. Pappus for 
■ onc ® displays a feeling for the heuristic motives of the older generation of 

geometers. For Apollonius and the olher authors of the analytic corpus, the ! 

discovery of solutions to problems was the heart of their geometric activity, and 

by their own account, the compilation of theorems was subordinate to this. But ! 

for the later generations, not sharing this heuristic motive, the power of the 

analytic method became merely superfluous—impressive, but no longer valuable. ' 


"... AND MANY AND THE GREATEST SOUGHT, BUT 
DID NOT FIND.” 

The search for solutions to the three problems of cube duplication, angle trisec¬ 
tion, and circle quadrature spanned the entire period of the classical geometry 
from Hippocrates to Apollonius and retained great interest for commentators 
throughout the later period of antiquity. Did the ancients view this effort as 
having succeeded in attaining its goal? 

In the cases of the cube duplication and the angle trisection, the commentators 
seent pleased that the solutions put forward answer to the nature of the problems. 
Hero and Pappus designate them as “solid** problems, in view of the successful 
application of conic sections toward their solution. Of course, this need not of 
itself rule out the possibility of finding a more elementary "planar” method. 
But Pappus’ almost abusive treatment of a certain colleague “pretending to be 
a great geometer" for his alleged solution of the cube duplication by planar 
means indicates that, to him, the book on this question would appear to have 
been closed. 1 " By analogy, we might expect that the several known constructions 
of the circle quadrature would leave him content to class the problem as "linear" 
and so to cease any search for a more elementary method. Indeed, he does just 
this in the case of the arbitrary division of the angle, classifying it as "linear" 
without a word on the possibility or advisability of looking for solutions other 
than the ones via the quadratrix and the spiral which he presents."* But he never 
actually makes this move for the circle quadrature, so that he may well have 
wished to keep the question open. At least one of the ancient solutions, the 
fallacious attempt by Bryson to equate the circle with a rectilinear figure somehow 
intermediate between rectilinear figures (squares?) drawn around and within the 
circle, suggests the aim to find a solution via the familiar elementary methods.'" 
Is this indeed the geometers’ objective in their research on this problem? The 
theme of the circle quadrature surfaces on several occasions in the discussions 
of the Aristotelian commentators, thanks to a few passing references to the 
problem in Aristotle’s Categories (Ch. 7). Posterior Analytics (I. Chs. 9 and 
12), and Physics (I. Ch. 2). Despite the nontechnical character of these sources, 
the commentators often avail themselves of authorities from the technical lit¬ 
erature as well as from the historical and philosophical writings on geometry. 
Thus, one might expect that they would reflect whatever conclusions had been 
drawn concerning the status of the circle quadrature among the ancient geometers. 
In this indirect way. then, they arc potential witnesses to the progress of these 
studies. 

Philoponus, as we have noted, makes the important distinction between the 
existence of the square equal to the circle and its actual construction; geometers 
assume the former, lie maintains, as the basis of their investigation of the latter. "* 
A similar observation is made by Eutocius in connection with Archimedes' 
theorem on the area of the circle. In proving that the circle equals the right 
triangle whose legs equal, respectively, the radius and the circumference of the 




362 Ancient Tradition of Geometric Problems 


Appraisal of the Analytic Field in Antiquity 363 


circle, has he not assumed that one can lake a line equal to the circumference, 
a thing which has been proved neither by him nor by anyone else? 

But one must immediately observe that Archimedes writes nothing outside of the 
things which arc appropriate. For it is somehow clear to everyone that the circum¬ 
ference of the circle is some magnitude* I Itclicvc, and this is among those extended 
in one |sc. dimension!, while the straight line is of the same kind. Even if it seemed 
not yet possible to produce a straight line equal to the circumference of the circle, 
nevertheless, the fact that there exists some straight line by nature equal to it is 
deemed by no one to Ire a matter for investigation. 1 Ir 

'Ibis is explicit admission of an appeal to intuitions of continuous magnitude, 
like several we have already cited, where the actual const met ion of the entity 
required is sharply separated from the issue of its existence."" Speaking as a 
mathematical commentator, Butocius is clearly of the view that no satisfactory 
construction of the solution to the circle quadrature has yet been achieved. Such 
is also the view of the philosophical commentator Ainmonius, disciple of Proclus 
and mentor of Butocius and Philoponus. In his remarks on Aristotle’s distinction 
between potential and actual knowledge (Categories, Ch. 7), he observes: 

The geometers, on constructing the square equal to the given rectilinear figure, 
sought whether it was possible to find a square equal to the given circle. And many 
and the greatest (of thcin| sought, but did not lind it. Only the divine Archimedes 
found an extremely good approximation, but the exact construction has not been 
found to this day. And this is perhaps impossible; for on these grounds even he 
(sc. Aristotle) has said “if indeed it is knowable iepistitonh ”—And perhaps for 
this reason he has made the straight line not dissimilar to the circular arc, if it is 
knowable or if it is not.—-Then he says that |if indeed| the quadrature of the circle 
is knowable, although no manner of its knowledge yet exists, then also from this 
what is knowable is prior to knowledge. M * 

The remark is reproduced with further elaborations by Philoponus and by Sim¬ 
plicius. 120 From this it would appear that the commentators leave the issue open: 
the search for a construction is valid, that is, it is in principle possible, even if 
no successful result has yet been achieved. In maintaining the latter, Ainmonius 
might seem merely to be ignorant of the more advanced constructing efforts, 
for he mentions only Archimedes' derivation of the estimate (3 l A) in the Di¬ 
mension of the Circle. But the same view, as we have seen, is held by Butocius, 
who can hardly be charged with such ignorance. Simplicius too is of this opinion, 
while at the same time knows through a report derived from lamblichus of the 
constructions via special curves proposed by Archimedes, Nicomedcs, Apollon¬ 
ius, and Carpus. 111 

The same distinction between the known and the knowable appears in the 
commentary on Buclid's Data by Marinus, a disciple of Proclus, active late in 
the 5th century a d. In specifying the variety of senses in which terms may be 
spoken of as "given,” Marinus introduces the class of "produceablcs” (porima), 
namely, those entities whose construction can in fact be presented. 122 Contrary 
to these are the "nonpmdttceablcs” (apora), of which the circle quadrature is 


an example. But Marinus goes on to distinguish two sorts of entities in this latter 
class: those not yet constructed, but capable of construction (these are termed 
poriston alios, "produceablc in the special sense,” as contrasted with those in 
the former class, termed kyrios porimon , "produced in the strict sense”); and 
the apt*ra, opposed to the porima, whose construciibility (tetesis) has not yet 
been decided (adiakritos). Alternatively, he terms the circle quadrature “un¬ 
known" (agnoston) yet "ordered” (or "fixed,” as it were of constant value— 
tetagmrnon). However pedantic this straining over terminological niceties might 
seem, the distinction Marinus is here striving to make between what is unknown 
and what is unknowable is an important one. Despite the long history of incon¬ 
clusive efforts on (lie circle quadrature, the ancient mathematical tradition re¬ 
mained unwilling to pronounce on the impossibility in principle of finding a 
solution. 

Closely related to this is the question of whether the diameter and the cir¬ 
cumference of the circle arc commensurable with each other. The development 
of geometric and arithmetic techniques for deciding on the rationality or irra¬ 
tionality of given terms, such as the incommensurability of the side and diameter 
of the square, was an important factor underlying the rigorizing movement within 
the pre-Buelidcan geometry. 12 ' Nevertheless, Greek geometers might sometimes 
assert a conclusion of this kind without the firm support of a demonstration. 
Consider Hero’s treatment of the problem of cutting off a segment equal to one- 
third a given circle (Metrica III, 18).'” He explains that he will give only a 
convenient approximation to the solution, “since it is clear that the problem is 
not rational." That his solution is not exact is plain; for by finding two chords 
drawn from a point on the circumference, he shows that the area lying between 
the cl amis is exactly one-third of the circle, so that a small additional segment 
must he neglected in viewing this to be a solution to the stated problem. But on 
what grounds I lero can conclude the irrationality of the problem we learn nothing 
more. In the matter of the irrationality of tt, comparable statements arc made 
by medieval and Renaissance writers like Mainionidcs, Maurolico, Stifel, and 
Gregory, all long before the first actual proof of this result was obtained by 
Lamliert in the 18th century. 1 ” Might these conjectures have been based on 
ancient precedents? Another remark from Butocius’ commentary on the Dimen¬ 
sion of the Circle is suggestive. In defending Archimedes against the charge 
raised by some critics that his values for the ratio were only rough approxima¬ 
tions, as compared with the closer values worked out by Apollonius and Ptolemy, 
Butocius writes; 

But if anyone felt strongly about effecting (his in minute detail, he could have 
used what is said in the Mathematical Syntaxis of Claudius Ptolemy.. , and I 
myself would have done this, save that 1 had in mind, as I have said many limes, 
that it is not possible exactly to find the straight line equal to the circumference 
of the circle by the things said here. 1 * 

Given that Butocius earlier observes, quite corccclly, that the square root of a 
nonsqnare number "cannot be found exactly,” 127 his reader might well be led 






364 Ancient Tradition of Geometric Problems 


Appraisal of the Analytic Field in Antiquity 365 


to infer that Eutocius likewise maintained the irrationality of the circumference 
and diameter of the circle, even if he does not flatly say that. Eutocius himself 
probably intends something weaker: that by qualifying his statement with the 
phrase “by the things said here,” he means only to claim that the specific 
pnx'edure set out by Archimedes cannot yield an exact value, since among other 
things it requires a scries of root approximations. Doubtless, within the long 
tradition of practical computation the constant recourse to approximations for 
the measurement of circles and similar figures would occasion surmises con¬ 
cerning their irrationality. But the ancient geometers did indeed make the essential 
distinction between experience and proof in this instance, as is revealed in this 
comment by Eutocius* contemporary, Simplicius: 

The reason why one still investigates the quadrature of the circle and the question 
as to whether there is a line equal to the circumference, despite their having 
remained entirely unsolved up to now, is the fact that no one has found nut that 
these are impossible either, in contrast with the incommensurability of tltc diameter 
and the side (of the square). 17 * 

In view of this firm .statement recognizing the need for a demonstration, not yet 
at hand, of the impossibility of solving these problems, we may infer that 
Simplicius maintained the same caution relative to the question of (heir irra¬ 
tionality. On these matters the ancient tradition thus admitted that the goal of 
their search remained elusive. 

What wc never learn from the ancient writers on the circle quadrature, how¬ 
ever, from the time of Aristotle and Eudcmus in the 4th century n.c. right through 
to the time of the commentators in the 5lli and 6th centuries a. tv, is what the 
precise constraints on a construction would be for any solution of this problem 
to be judged acceptable. Rut certain responses to the constructions which were 
proposed offer us some insights. Speaking of the methods used hy Archimedes 
and others, Simplicius remarks that “it would appear that they all made the 
construction of the theorem mechanical {organike)", 129 In the same context, but 
now responding to a claim made by Porphyry (that “most learned” nco-Platonisl 
commentator of the 3rd century a. tv), he remarks that “perhaps a certain me¬ 
chanical discovery of the theorem had been made, but not a demonstrative 
(apodeiktiki) one”. ,,vl Simplicius appears to maintain that a merely mechanical 
construction, corresponding to the use of curves of the “linear” class, does not 
satisfy the conditions for a geometric demonstration. In this, he conforms to ail 
attitude already expressed hy Pappus in his reservations over the “rather me¬ 
chanical manner of generation** of the quadratrix, for which he can find a 
precedent in the criticisms made earlier by Spools. 131 A similar view arises 
among the Arabic geometers, who distinguish between the “mobile** and the 
“fixed” sorts of geometry, doubtless by way of transferring the ancients* dis¬ 
tinction between “mechanical** and *‘demonstrative,’* and they too manifest a 
certain preference for the latter. 1,1 Wc have already seen how Plutarch, a thinker 
with strong Platonist leanings, cast Plato himself in the role of criticizing the 
supposedly mechanical methods of Eudoxus and Menacchtmis for solvittg the 


cube duplication, and he even presumes to impute to Archimedes an antime¬ 
chanical outlook on (he field of geometry. 133 But the later commentators are 
hardly unanimous in adopting this biased attitude. Gcminiis and Carpus ac¬ 
knowledge the great importance of mechanical pursuits in the mathematical work 
of Archimedes, even if his writings stress the precision of geometric theory. 1 ' 4 
In offering a mechanical method of his own for solving the cube duplication. 
Pappus shares in the spirit of the geometers of the 3rd century n.c., like Era¬ 
tosthenes, Nicoinedcs and Diodes, whose search for practical solutions to such 
problems led them to explore mechanical approaches. 135 Both Hero and Pappus 
wrote extensively on the application of geometric techniques in mechanics. It is 
such a context which occasions Hero’s presentation of the neusis for the cube 
duplication, for instance, and one readily detects the practical interest attached 
to his accounts of the measurement and section of figures in the Metrical 
When Pappus writes that * ‘those problems in mechanics called ’organic* arise 
through being separated from the geometric domain,” his moaning might not 
he entirely clear. ,3, But it becomes so when we consider the item for which this 
serves as preface: the solution of the problem of gauging the diameter of a 
mutilated cylindrical column; for this leads, through a series of lemmas on 
ellipses, to the determination of the ellipse passing through five given points. 1 '" 
This is but one of many instances where the ancient geometers found in me¬ 
chanical situations a source of interesting problems for geometric research. 

When commentators like Simplicius voice their dissatisfaction with such 
mechanical approaches to the circle quadrature, they still do not clearly say 
what the alternative “demonstrative” or “geometric” construction ought lo 
consist of. Presumably, their hope was for an elementary planar solution, or 
even a solid one. But if efforts along these lines actually were launched, 
evidence of them lias not survived. To the extent that they insist on some 
such method, they also reveal an insensitivity to the origins of geometric 
constructions ami means of construction; for how are the “planar” techniques 
of circles and straight lines any other than mechanical, that is, derived by 
way of abstraction from the practical use of compasses and rulers? 1 ' 9 Simi¬ 
larly, the conics arc no more or less mechanical than the curve formed by 
Archytas through the sectioning of solids. It is only through an arbitrary 
process of fiat by postulate that Euclid transforms these mechanical means 
into the starting points of a geometric theory. But for the conics and the field 
of solid geometry, even this process remains incomplete; for neither Euclid 
nor Apollonius eliminates the appeal to mechanical conceptions, specifically 
the rotations of plane figures about fixed axes, for the generation of cones 
and other solid figures, nor docs either ever specify what the process of 
sectioning solids like these involves, whether hy postulate or by theorem.* 4 ' 1 
Thai the process is not merely conceptual is evident from the discussions of 
solid figures by Hero and Pappus, mentioned just above. 

The situation relative to the constructing technique via neusis is the same. 
In some instances, as in Ihe cube duplications of Philo, Hero, and Pappus, 
an actual ruler (kanonion) is conceived as being manipulated about a pivot 





jw Aficictit i rauii ion ui ucomcmc rrooicim 


'♦it 


until the desired configuration is achieved.' 41 But in other instances, the 
operation is expressed in abstract terms: to draw a line such that, while 
inclining ( neuousa ) toward a given point, it marks off segments having a 
specified property. Such is the mode adopted by Hippocrates (in the account 
from Eudemus) for the construction of the third lunule, by Archimedes in the 
lemmas to the theorems on spirals, and by Sporus in his cube duplication (an 
alternative equivalent to the manner followed by Dioclcs and Pappus).' 41 Such 
also is the manner of the cube duplication of Apollonius, as reported by 
Philoponus; it is nothing other than the abstract equivalent of the ruler con¬ 
struction given by Philo. Where the condition of the inclining line enters, 
one reads, "but this is assumed as a postulate ( aitema ) without proof ( ana- 
podeikton).'"*' Now, this is surely a strategy one might have adopted: to 
posit a few of the generalized configurations of neuses as formal postulates 
and from this begin to elaborate a sequence of problems and theorems. But 
the ancients never seem to have taken this approach. For Apollonius and his 
colleagues, neuses were treated as problems to be effected in some instances, 
as for the cube duplication here or for the angle trisection, by means of conic 
sections, while in other instances by planar means.' 44 Similarly, the motion- 
generated "linear" curves, like the spirals, conchoids and quadratrices, each 
have their own "primitive property" (archaikon symptdma)\ ; for instance, for 
the plane spiral it is that the rays drawn to the curve from the origin have the 
same ratio as the angles they make with the initial ray. All further properties 
and applications of these curves arc effected as deductions from their symp- 
tomata. If these curves are naturally conceived through mechanical motions, 
they need not have been treated as such in the formal exposition of their 
properties. That this formaliz.ation was not done, with the result that these 
curves came to occupy a subordinate position in the hierarchy of construction 
techniques, is thus largely accidental, the fault of their relative unfainiliarity, 
their lateness of invention, their more advanced character, or their specialized 
range of applicability, by comparison with the constructions presented by 
Euclid and Apollonius. 

The central issue still eludes us: the sheer diversity of solutions proposed 
for each of the three special problems would indicate that the ancient geo¬ 
meters were engaged in a search never consummated to their satisfaction. 
What was the object of that search? The response, that they sought planar 
solutions for these problems, attempts to provide a simple motive behind the 
ancient geometry. It is a view which the commentators in late antiquity appear 
to maintain, as do many—indeed, most—modern scholars who write on this 
subject. But it is surely a brutal oversimplification; for in trying to probe the 
motives of researchers, we face an issue which has psychological aspects as 
well as formal mathematical ones, and this is likely to require a more complex 
account. Part of the answer, I believe, lies in recognizing a certain symbiosis 
between problems and the methods for solving them. The activity of studying 
and solving (or even not solving) problems gives rise to new techniques; the 
latter in their tum may gain a life of their own, and through further elaboration 


give rise to new problems and new techniques. For instance, Archimedes' 
neusis for the angle trisection could suggest to Nicomcdes the mechanical 
mode for generating the conchoids; but then, in addition to the initiating 
context of the problem, the curves draw attention to their asymptotic behavior 
and other properties and to the possibility of defining related curves, like the 
circle-based conchoids, and examining the applications of these. The later 
commentators do not sense this flexible interaction between problems and 
methods, this participation in a heuristic process, but instead attempt to de¬ 
scribe it in terms of a specific product, the goal of research. But their inability 
actually to articulate this goal, to state precisely what form of construction 
was being sought within the earlier geometric tradition, reveals that no such 
specific goal operated as the motivating clement for those geometers. 

EPILOGUE 

The Greek historical writer Herodotus tells the story of how Pharaoh Psam- 
metichus set out to determine which of the races of man was the oldest.' 4 ’ 
He had two newborn babes carried off to be reared in isolation from society 
by a herdsman who was to report back as soon as they started uttering rec¬ 
ognizable words. In the meantime he was under orders not to speak in their 
presence, but only to see to their physical needs, such as bringing in the goats 
every now and again to provide them milk. After about two years the herdsman 
noticed that the children would say bekos whenever he came to look in on 
them. He duly informed the king who found through further investigation 
that bekos was the Phrygian word for bread. For this reason, Herodotus says, 
the Egyptians from that day onward acknowledged the Phrygians as an older 
race than their own. 

However much one might hope to discern in this account a prototype of 
the experimental method in the social sciences, it is clear that Herodotus is 
joking. Hie children were not speaking Phrygian, they were speaking goat. 
The ancient commentators on the analytic tradition of geometry slipped in 
much the same way. For the Aristotelian commentators, as also for the Neo- 
platonist Proclus. philosophical questions were naturally the principal interest. 
Geminus, significant for his encyclopedic survey of mathematics, was an 
adherent of the Stoic philosophy. Through him and his like, other writers 
such as Hero and Pappus, while engaged primarily in the presentation of 
mathematical texts, might thus import philosophical elements into their more 
general observations ou the nature of mathematics. In this way, the later 
tradition of commentators applied the language and concepts of philosophy 
in their effort to describe, interpret, and criticize the works of the older 
tradition of geometers. 

We have seen the fniits of their inquiry. Notions of the ideal nature of 
geometric entities lead them to emphasize the priority of theorems over prob¬ 
lems for the exposition of geometric truths. The method of analysis, an ef¬ 
fective instrument for discovering the solutions of problems, is purported to 





368 Ancient Tradition of Geometric Problems 


Appraisal of the Analytic l ; icld in Antiquity 369 


serve as well in the investigation of theorems, a claim little encouraged through 
a consideration of the actual literature of geometry. Others, interested in 
giving an account of the logical structure of geometry, assign to problems 
the role of justifying the assumptions on the existence of entities made in 
theorems. Stimulating as an insight into the logical account of geometry, this 
view captures neither the diversity and power of the problematic form, nor 
the methods actually adopted by geometers for addressing issues of existence 
when they arose in their theorems. What the early geometers introduced as 
a useful division of loci becomes, among the commentators, a normative 
division of problems in general, imposing in an unworkable form the rec¬ 
ommendation to seek planar methods of solution. At the same time, the rich 
field of mechanical methods is assigned to the periphery of the domain of 
geometry. In all these ways, the philosophical preoccupations of the com¬ 
mentators render them insensitive to the nature and motives of the older 
tradition of research. 

Much the same applies to more recent scholarship. Examining the ancient 
geometry under the conception that its objectives were primarily formal, for 
instance, or that it subscribed to a constructivist mctamathematical program 
or that it otherwise reflected awareness of concerns characteristic of later 
movements in mathematics and mathematical philosophy might yield an ac¬ 
count more sophisticated than that proposed by the commentators from late 
antiquity, but is not for this more likely to be faithful to the earlier research. 
Indeed, the modern accounts typically attempt to build on the start provided 
by the commentators, assigning greater weight to a single passage by Pappus 
on the nature of the method of analysis or to a single short comment on the 
relative ordering of the techniques of construction than to the whole corpus 
on the analysis of problems. Once it is recognized that such statements by 
the commentators are founded not on the inspection of and meditation upon 
geometrical materials, hut on paraphrases of slock positions on the related 
general questions discussed in the philosophical literature, one will know 
better how to approach the geometrical writers on the one hand and the 
commentators on the other in order to gain the appropriate insights into the 
issues which happen to be of interest. To be sure, the geometers were involved 
in a philosophically interesting effort, just as Herodotus* children were ut¬ 
tering sounds interprctablc as Phrygian words. But the ones were not engaging 
in philosophy, just as the others were not speaking in Phrygian. Inferences 
on such a basis as to the relative significance of philosophical considerations 
within the ancient geometry will be no more valid than the Egyptians’ de¬ 
ductions on the relative antiquity of their race. 

The present survey of the ancient geometry has called attention to the 
prominence of the activity of investigating problems of construction. Per¬ 
ceiving that the problems of circle quadrature and of cube duplication did not 
lend themselves to solutions comparable to those more elementary problems, 
like the rectangle quadrature or the interpolation of a single mean proportional, 
effectihlc via familiar techniques like the use of circles and straight lines, 


Hippocrates advanced the study of both, the former through his quadrature 
of the lunttles, the latter by reducing it to the finding of two mean propor¬ 
tionals. Subsequent efforts by Eudoxus and his disciples developed new geo¬ 
metric techniques in the context of these and related problems: the generation 
of special curves, for instance; the implementation of the theory of the ap¬ 
plication of areas, later serving as a pillar of the theory of conic sections; the 
limiting techniques appropriate for effecting the measurement of curvilinear 
figures; and the versatile method of analysis, to become the most characteristic 
feature of the later problem-solving effort. 

In the 3rd century B c. this activity undergoes a split, maintained with 
remarkably little overlap. One branch, centering on the circle quadrature and 
related inquiries, with the concomitant emphasis on the use of the Eudoxean 
limiting methods, is dominated by the discoveries of Archimedes, who not 
only applied these methods successfully to new figures, like the conoids of 
revolution, hut also effected technical refinements on the methods themselves. 
His findings by means of neusis constructions and his studies of the spiral, 
a model instance for the class of curves generated via mechanical motions, 
inspired several of the geometers in the following generation in their search 
for alternative constructions of the three ’‘classical” problems. 

Hut a second branch of the ptoblcm-solving activity during this period 
developed in a manner little influenced by the work of Archimedes. Spurred 
by the studies of locus and the applications of the theory of conics initiated 
by the work of Euclid and his contemporaries, geometers extended the domain 
of problems effected via the method of analysis. By the time of Apollonius, 
about a century later, the wealth of results permitted one to gain a sense of 
the structure of the field: which forms of conditions give rise to planar loci 
(circles and lines), which others to solid loci (conic sections), and accordingly, 
which kinds of problems arc amenable to planar or solid solving methods. 
The surviving evidence, even in its highly imperfect condition, makes clear 
that these geometers aimed at exhaustion in their researches into specific 
classes of problems. For instance, Apollonius in one treatise secured the 
construction of circles answering to conditions of tangency over the whole 
range of possible orientations and devoted other works toward effecting the 
same for the conics. He explored in detail the manner of effecting forms of 
neusis constructions via the planar methods, while he and other geometers 
pursued the investigation of solid problems like the solution of the three- and 
lour-linc locus. 

Throughout this two-ccntury span of research, the preeminent aim is toward 
finding solutions to problems and discovering new properties of geometric 
figures. I’hc geometers themselves stress this heuristic goal: Archimedes de¬ 
scribes with evident pride his “mechanical method** for finding the measures 
of surfaces and solids and centers of gravity; Apollonius presents his com¬ 
pendium on the conics as an instrument for investigating locus problems; even 
the commentator Pappus acknowledges that the corpus of analysis is organized 
wholly for abetting the solution of problems. This ideal, even without such 








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372 Ancient Tradition of Geometric Problems 

to base an inference, however, since such cited titles need not be original to the works. 
Proclus reports of the geometer Hennotimus of Colophon, shortly before the time of 
Euclid, that in addition to discovering many of the things in the Elements, lie “compiled 
certain things on the loci’* </« Eui'lidem, p. 67). This might give Herniotiimis a role in 
the solution of problems like that in the Meteorologica III, 5; but again, we cannot infer 
from Proclus* report that the term "locus** was already then used to designate such 
problems. It is of interest to note the series of theorems from Euclid's Optics, Props. 37, 
38, 44-47, of which the following is representative: “There is a place {topos) where, 
when the eye is fixed while the object seen is moved, the latter will always appear equal 
(in size)** {ibid., 37; note the differences in the Hiconine versions. Props. 41-47). These 
theorems all deal with the specification of topoi where certain optical phenomena obtain. 
But at the same time they yield a geometrical figure corresponding to each topos. It seems 
to me possible that contexts of this son in optics, where topos retains its basic sense of 
“physical place/' served as background for the early study of geometrical locus. This 
would be but one of several ways in which optical studies contributed to new developments 
within (he geometric field in antiquity (cf. the discussion of the early theory of conics in 
Chapter 4; and Diodes* work on burning mirrors in Chapter 6), 

On these efforts by Mcnnechmus and Archimedes, see Chapters 3 ami 5. 

,T See the rdsumd by Heath, History , II, pp. 185-189, 

** This view is elaborated below. 

19 Collection (IV), I, pp. 270-272; cf. note 10 above. 

w See, for instance, IIcath, History, I, pp. 218 f, who is markedly more circumspect 
in this matter than most writers. The assumption of the quest lor planar constructions is 
carefully scrutinized and convincingly discredited by A. D. Steele, “Ueber die Rollc von 
Zirkcl mid Lineal/' Quellen and Studien, 1936, 3, pp, 289-369. 

11 On the problem from Apollonius cited here, see Chapter 7. 

'* ‘See Chapter 5(ii) for the solid construction of Archimedes* neusis. 

11 See Chapter 2. Steele provides a very thorough account of this construction, 
arguing that the neusis cannot have been intended to be effected in any way other than 
as a neusis; “Zirkel und Lineal/* pp. 319-322. 

* See Chapter 3. 

w See Chapter 4. 

* See Chapter 6. 

1 See Chapters 5 ami 6. Note that Archimedes' solution to the circle quadrature 
via drawing tangents to the spiral is not unrelated to the procedure id neusis. 

w Sec Chapter 7. 

w Such classifications can define the project of specifying the relative domains of 
problems solvable by the various constructing means. An impression of the wide range 
of results to be obtained in this area can be derived from the articles in Enriques' survey 
(cf. note 42). 

40 Pappus, like Hero, actually employs a neusis; hut his method is equivalent to 
Diodes* use of an auxiliary curve (sec Chapter 6). 

41 Sec the statement at the end of text A cited above, and his reproduction of Hero's 

».* ll*<* .. * f. ..a II 




Appraisal of the Analytic Field in Antiquity 373 


art. 7, pp. 193-195. The incducibilily of the third-order relations for the cube duplication 
and the angle triscction was lirst shown by L. Wantzel (1837); see Liouvilie’s Journal, 
2. pp. 366 -372. 

41 Collection, I. p. 30. His lengthy and unsympathetic account of the proposed 
method {ibid, pp. 32-48) should hegin to raise doubts among those who would assume 
in Pappus himself a high level of geometric expertise. See the brief summary by Heath, 
History, I, pp. 268 270. 

44 The Academic fran^aise ruled in 1775 against the further consideration of con¬ 
tributions (dating to the circle quadrature (cf. P. Beckmann, A History of Pi, 2nd ed., 
1971, p. 173); but the first successful proof of the transcendcntality of it came only with 
Lindemann's publication in 1882, extending an earlier result by Hermitc (1873) (cf. art. 
8 in Enriques, op. cit.). Similarly, Descartes pronounced that “it will be an error to 
strive in vain to wish to construct a problem by a class of lines simpler than its own 
nature permits” (GY ometrie, ed. Smith, p. 157) precisely two centuries before the first 
specific results on the rcducibility of the associated polynomials were presented (see note 
42). 

41 Collection (III), I, p. 56. 

** Cf. Elements 1, 22. The implied governing construction might be “it is pro¬ 
posed. .. ** iproballeud, or proteinetai), as sometimes supplied in discussions by Pappus, 
Proclus, and others, or alternatively “it is possible ... ' '{dutuiton esti ). Of course, the 
actual construction establishes far more than the mere possibility of the construction. 

41 Cf. Elements, I, 21. 

° Ibid., I, 20. 

w Cf. the locus problem in the MeteoroloRica III, 5 (Chaplcr 4 above). 

141 l : or an example of just this sort from Pappus, see Chapter 4, note 16. 

" Pappus, Collection (VII), II, pp. 648 ff; Proclus, In Euclidem, pp. 212, 302. 
Proclus recognizes an alternative sense of “porism/* namely “corollary** {ibid., pp. 
301-303) ’Hie term derives from ponzein, “to provide” or **lo devise.** An appearance 
of a form of the verb in the MeteoroloRica (III, 5. 376 a 15) may carry the technical 
sense, although another appearance in a mathematical passage in the Ethics (1112 b 26; 
discussed below) seems not to. 

42 In Euclidem, p. 81. 

41 litis is a view proposed, among others, by Proclus and by Zeuthen. See note 
77. 

M Note the propositions of the form “there arc places {topoi) such that.. .** in 
Euclid's Optics (see note 25 above), llicsc are undeniably existential, both in sense and 
in form. But they arc framed as theorems and serve as the basis for a problem (Prop. 
48): “to find the places where the equal magnitude appears half ns big, etc.” Similarly, 
the materials in the pseudo-Euclidean Catoptrica culminate in a problem (Prop. 29): “it 
is possible to construct a mirror such (hat many faces arc seen in it, some bigger, sonic 
smaller, some nearer, some further away, some reversed, some not/* This serves merely 
to bring together the various results already established on plane, convex, and concave 
mirrors; it is hardly the equivalent of an existence proof. 

44 Pappus, Collection (IV), I, pp. 272 ff. See Chapter 7 and my “Hyperbola- 






374 Ancient Tradition of Geometric Problems 

54 Conics, Preface to Book I (ed. Heiberg, I, p. 4): ’‘the third book (contains) 
many remarkable theorems useful for the syntheses and diorisms of solid loci. 1 * Similarly, 
the contents of Book IV are said to he "useful for the analyses of diorisms, 11 while Book 
V also is "needed for obtaining a knowledge of the analysis and determination of prob¬ 
lems, as well as for their synthesis* 1 (cf. Heath, Apollonius , p. Ixxiv). The same attitude 
is maintained by Pappus when he describes the topos analyomenos as "a certain special 
corpus ... conducive to the (hiding of problems ... and compiled as useful for this 
purpose alone 41 ( Collection , VII, Preface, ed. Ilultsch, II, p. 634). 

See Chapter 7. 

’* Archimedes does without any such existence proof in the area theorem of 
Dimension of the Circle, Prop. 1; the omission was criticized by some, but defended by 
Eutocius (Archimedes, Opera, 111, p. 230; sec discussion below). Further examples 
relating to the alleged existential function of problems appear in chapter 3. 

99 Note that it was the thcorcmatic form of Postulates 4 and 5 which encouraged 
ancient geometers like Apollonius and Ptolemy, respectively, to seek proofs for them; 
cf. Proclus, In Eudidem , pp, 188 ff, 191 ff, 365 fl. Proclus asserts llatly that the 
"postulate 11 on parallels is a theorem (ibid., p. 191). 

Collection (III), I, p. 30; cf. (VII), II, pp. 648 ff. Note that the notion of 
displacing all problems by theorems or vice versa receives no support from the technical 
literature, where both forms appear and serve clearly marked purposes. Comparable 
distinctions in the dialectical roles of problems and theorems (c.g., the distinction between 
problfmata and protaseis in debating contexts. Topics 1,4, 10, 11) or in the different 
priority of terms in the order of knowing and the order of proving (e.g. Posterior Analytics 
I, 2) may suggest a loose connection between these discussions on geometry and the 
Aristotelian epistemology. 

*' Republic 527; cf. 5!0d and Euthxdemus 290c. 

** In Eudidem, pp. 77 f. On Amphinomus, see Chapter 3. 

91 See the discussion of the Plautnicus fragment. Chapter 2. 

M in Eudidem, p. 243. 

M Ibid., p. 78. 

M See Aristotle's Metaphysics Z. 10, 1036 a 9; and the discussion by Heath, 
Mathematics in Aristotle, pp. 213-216, 224-226. 

** Proclus. in Eudidem , pp. 241 f; Pappus, Collection (VIII). Ill, p. 1026; Sim¬ 
plicius. in Physica, ed. Diels, p. 60. The identity of this curve is not known; Tannery 
conjectures it was the cycloid, but I have argued that it may he the cylindrical helix (sec 
my "Archimedes and the Spirals, 11 pp. 73 0. 

M in Eudidem , p. 242. 

~ Ibid., pp. 80 f. 

*' Here Zenodntus is described as "in the succession of Ocnopidcs 11 and "among 
the disciples of Andron.** While Oenopidcs would be the 5th-ccntury geometer-astron¬ 
omer, this helps little in placing Zenodotus. Given the later reference to Posidonius in 
this same passage, an identification with a Stoic writer of this name from the 2nd century 
R c. might be indicated fcf. Pauly Wissowa, "Zenodotus (4) and (6)"), hut this is entirely 
conjectural. So also is the degree to which such late writers reflect the actual ideas of 
the earlier writers with whom they arc associated. K. von Fritz has observed that Zen- 


Appraisal of the Analytic Field in Antiquity 375 

odotus 1 observation about problems (i.e., that they seek "that on the assumption of which 
something is, 11 tinos ontos ti esti) has to do with the analysis of problems ("Ocnopidcs** 
in Pauly Wissoyva, reprinted in his Schriften zur griechischen Logik, Stuttgart, 1978, 2, 
pp. 151 f). Note, however, that the implied conception of analysis accords with the view 
of Pappus, following the Aristotelian manner, that is. as a search for suitable antecedents, 
rather than as an investigation of consequences (see discussion below). Of course, one 
can hardly hope to ascribe to Ocnopidcs a role in the introduction of the analytic method 
on the basis of this evidence. 

” Tliis is surely Posidonius, the distinguished Stoic teacher of the late 2nd century 

II c 

99 The manuscripts here insert probUma which Morrow understandably rejects as 
an interpolation, since it violates the sense of the passage (cf. his Proclus A Commen* 
tary. . .. p 66n). 

M In Eudulem, pp. 233 f 

’* A view minimizing the formal intent underlying Euclid's geometry is argued 
by A. Seidenherg in "Did Euclid's Elements, Book I, Develop Geometry Axiomatical ly?" 
Archive for History of Exact Sciences, 14, 1975, pp. 263-295. 

” Oddly, Proclus can here cite some Epicureans who criticize proving this result 
at all, given its perfectly obvious character (ibid., p. 322). 

* Sec Heath, Euclid's Elements, 1 , pp. 293 f. Note (hat if the analysis of this 
problem had been given first, the necessity of this condition and its role within the synthesis 
would have been clear. It would so apjtcar that Euclid’s formal difficulties in eases like 
this follow from his decision to present only the syntheses of the problems. 

” I believe (hat the locus dassicus for the thesis that the ancient geometric con¬ 
structions were meant to serve as proofs of existence is Zeuthen's "Die geomctrischc 
Onstiitction als 'Existciizhcwcis* in der antiken Geometric, 11 Mathematische Annaten, 
47, IK96. pp. 222-228. Ilis case rests primarily on Euclidean materials, although he 
draws from more advanced work as well. But when he supposes, for instance, that the 
(necks had lo seek a construction of the conics (c.g., as sections of a cone) because they 
would not admit mechanical or pointwisc methods, he commits a plain error (cf. the 
discussions of Mrnacchmtis and Dioclcs in Chapters 3 and 6). It is thus hard to understand 
how the thesis has gained such credence. To be sure, one can find telling instances where 
problems enter hi justify flic introduction of terms necessary for subsequent theorems. 
An instance not cited by Zeuthen is Elements XII. 16, 17: two problems assumed in steps 
in the proof of Prop. 18 (that spheres are as the cubes of their diameters). But we have 
seen other instances where existence is established via theorems, subordinate to problems 
(cf. note 54 above), as well as many cases where the problems of construction have 
standing in their own right, without reference to associated theorems (cf. note 56 and 
Chapter 3). We can only conclude that the ancients did not distinguish proofs of existence 
from other geometric propositions in the way modem mathematicians do; that apart from 
diorisms (which arc by their nature existential in force) no special format was reserved 
for questions of existence as they arose, these being handled now as theorems, now as 
problems, but often only as tacit assumptions. For a nicely balanced view on the relation 
of ancient geometric constnictions anti existence proofs, see 1. Mueller, Philosophy of 
Mathematics . pp. 15 f, 27-29. 

” Iw the scholia, sec Euclid, Opera, ed. Heiberg. IV, pp. 364 ff. 'Flic commentary 
by at-Nairi/l (Annarilius) survives both in Arabic and in its tatin translation by Gerard 









376 Ancient Tradition of Geometric Problems 

of Cremona. I have consulted the latter in the edition by M. Cuctze, in Euclid, Opera , 
ed. Heiberg, IX; for references to the former, sec P Sezgin, Geschichte ties arabischen 
Schrif(turns, V, pp. 283-285. 

" Sec the contributions cited by J. Hinlikka and l1. Heines, 77ir Method of Analysis, 
1974, Ch. ii. The authors have compiled an extensive survey of ancient comments on 
analysis in their Ch. viii. 

Collection (VII), II, p. 634. 

11 Ibid. 

91 Euclid, Opera, IV, p. 364. 

*' I follow the Latin of Gerard in ibid., IX, p. 89. He seems here to have made 
a valiant attempt to render a difficult passage. The Arabic (Leiden, Or. 399, f. 25v. lin. 
10—12) can be given thus: 

As for analysis (tahlil). it is when any question has Ivcn proposed h» us. we say (lhaO we 
lake it down ( ntmztiuha ) (to) the degree (mrrnjj/ir) of the thing sought (nut(lub), that il is 
found Unruihu rtutwjud). ihen we break it up (nafiuiifuhu) to something whose proof has 
already preceded, so that since it is manifest to us. we say that what was sought [motlub) 
has now l>ccn found (tru/ti/rj) through analysis. 

Hinlikka and Remes cite a translation made for them by T. Harviainen from the Arabic; 
“we suppose (he thing sought as being imawjud), then we resolve it to something already 
proved.*' [Method of Analysis, p. 93) They thus wish to read iruuthu nutwjtid as a version 
of Greek hos on. But in context, mawjud must surely here mean “found,** as the verbal 
form wujida clearly does later in the passage. The underlying Greek would thus he hris 
zftoumetwn, that is, “as if found.** 'Hie Arabic translator has lost the nuance of lun 
(present in the parallel Greek passages cited; hOs homologoumenon). Consequently, Ger¬ 
ard renders as que est inventn what, in light of the presumed Greek, would more accurately 
be quasi inventa. As for terminology, “analysis” becomes talflii C'dissolutio"), * ‘syn¬ 
thesis** is tarkib {"compositin''). 

M This looseness of the senses of akolouthein is emphasized by Hinlikka ami 
Kernes, op. cit ., pp. 13 ff; they opt for the rendering “concomitants.** 

M Note that (he scholiast speaks of the thing admitted as being “true," and so 
would seem to have in mind a thcorematic context of analysis, like that described later 
by Pappus. Hero's term “known** could apply equally well to the analysis of problems 
and that of theorems. The phrase “up to something admitted as true** here only poorly 
suits the context, and is likely to be a scribal emir due to the appearance of the same 
phrase in the preceding sentence (see note 82). Some manuscripts here have an alternative 
phrase, “up to the ending up nr taking of what is sought” (cf Heiberg, Etuiulis Opera, 
IV, p. 366n). 

** It seems difficult to assign to *‘antecedents** and “consequents” here their usual 
logical senses and yet to remain compatible with Pappus' notion of the order of analysis 
given before; for there the terms found were logical antecedents, even though they 
happened to he discovered later in the analysis. Of course, in the usual presentation of 
an analysis, terms are derived as logical consequences, so that a reversal of the logical 
order docs indeed have to he secured in producing the synthesis. 

In the Metrica, Hero presents many examples of “analyses** of an interesting 
variant sort. In these, the objective is to derive a computational rule or procedure by 


Appraisal of the Analytic Field in Antiquity 377 


means of what is called an “analysis** and then in (he following “synthesis** to work 
out a solution in particular numbers via the derived rule. In Metrica III, 4-14, 20-22, 
these arc all problems in the division of figures, and one can appreciate (he appropriateness 
of the analysis/synlhcsis distinction. But in Metrica I, 8-25 and II, 6-9, 13, the rules all 
involve expressions for the atcas and volumes of figutes. These arc all staled ns problems 
(e g., “to find the area of the regular heptagon*’ in I. 20), but they have no analytic 
hypothesis of the son which dominates the analysis of a problem. On the other hand, 
these analyses do start from the whole figure ami work through to a relation of a computable 
sort pertaining to the givens te.g., the given length of 10 for the side of the heptagon), 
while the syntheses begin with the givens and calculate with these in accordance with 
the derived relation to produce what is sought. Doubtless, this formal was adapted from 
the analysis of problems in the more advanced geometry. llero*s version may be based 
on a prior tradition following this same format; he depends, for instance, on values drawn 
from “the books on chords’* (a reference to Hipparchus? cf. Opera , ed. Schdne, p. 58 
n), but his heading to the section of analyses and syntheses is ambiguous (ibid., p. 16). 
At any rate. Pappus seems to know of this formal; for he invites the reader to work out 
numerically a value for the cube duplication “in accordance with the analysis*' through 
consultation of Ptolemy’s table of chords ( Collection , I. p. 48). 

"* Collection (VII), II, pp. 708-710; cf. Heath, History, 11, pp. 406 f. 

" v See. for instance, analyses of the problems on the inscription of the regular 
polyhedrn in Collection III, Prop. 54 ff. ’Hie same appears in Diodes’ analysis of Ar¬ 
chimedes' solid problem (cf, ‘loonier. Diodes, p. 86). In some instances, the synthesis 
might merely he omitted without comment, as in Pappus' treatment of the analyses of 
the “solid nrinu” used by Archimedes {Collection (IV), I, Prop. 42 f). Il is interesting, 
however, that hi most such eases the commentators Eutocius and Pappus will supply the 
missing synthesis, or at least fill in a sketch of it. 

It is of course possible that Pappus himself is responsible for this lemma. Il 
would then merely he a reflection of his own view of the nature of analysis, rather than 
evidence from the more general technical field. 

Nicomachean Ethics II 1.3, 1112b 15-27. 

** Hie passage is well noted by Hintikka and Remes (op. cit., pp. 85 0 who draw 
attention to the rcmaikahlc conformity in terminology between it and the account of 
analysis by Pappus. Bin they seem reluctant to admit that Pappus might have referred to 
it as a model for his own statement. 

lliis observation is made by Hintikka and Remes, op. cit., p. 86. 

w Collection (III), I. p. 86 (i c . that the geometric mean is a cause of harmony 
in all things; this would appear to relate to Timaeus 31-32, as noted by llultsch, ibid., 
p. K7n). 

" Hicsc arc noted by Thomson in his introduction to the Commentary of Pop* 
pus..., pp. 51-57. 

** An extreme example of the mathematical commentary conceived within the 
framcwoilc of Plalonist doctrines is that by Proclus on Euclid's Book I; see the introductory 
remarks by Morrow, op. cit., pp. xxiv-xliii. 

Collection (VII), II, pp. 634-636. 

M Note how Pappus' view here of the two passible outcomes of the analysis recalls 
Aristotle's observation to the same effect in the Ethics passage cited above. 







378 Ancient Tradition of Geometric Problems 

w Cf. Hultsch in Collection , II, p. 636n. This line is deleted without comment in 
the versions of Pappus' account given by Heath ( History , II, p. 401) and by I. Thomas 
(Greek Mathematical Works , II, p. 598). 

"" Op . n7., Chs. iv-vi. 

>oi Hintikka and Kcmcs (ibid.. Ch. iv) insist that the theoretic form of analysis he 
taken quite seriously and attack in sharp terms any view that the heuristic processes 
underlying problem solving and theorem proving arc somehow distinct (ibid, , p. 47. 
criticizing M. Mahoney, "Another t^ok at Greek Geometrical Analysis," Archive for 
History of Exact Sciences, 1968-69, 5, pp. 319-348, especially p. 328). But surely two 
different things arc at issue here. Let us of course accept the authors* findings on the 
logical description of the problems and theorems, to the effect that the description is the 
same in either case. Nevertheless, the analysis of a theorem in the sense they require 
must begin with the hypothesis (hat the result claimed in the theorem is true. How is that 
result arrived at? In (he pedagogical context, the teacher might present a claimed theorem 
to the student and ask him to prove (or disprove) it; in this case, the student might well 
employ an analytic procedure to assist him in working out a proof. Otherwise, the analytic 
hypothesis must he a matter of mere conjecture. One can well doubt whether the ancients 
approached the linding of proofs of theorems as theorems in such a context-free manner. 
Furthermore, in cases like Archimedes* theorem on the area of the circle (that it equals 
one-half the radius times the circumference), the proof results from hypothesizing that 
the claimed result is false . It is hard to see the value, from the historical point of view, 
of assimilating the techniques used for investigating such theorems with the analytic 
method used for problems, even if this proves to be rewarding from the logical point of 
view. 

m Collection (IV), I, p. 186. This example is discussed in detail in Hintikka and 
Remes, op. ciV., Ch. iii. The lemma which closes Collection VII (II, pp. 1016-20) is 
also an attempt toward the analysis of the theorem, although the text has been severely 
garbled. Other than these, I know of no extant analyses of theorems, save for those of 
the trivial sort mentioned earlier (see note 78 and note 87). 

,m Cf. Prior Analytics I, 23, 41 a 21 If. 

104 Cf. Proclus. In Euclidem. pp. 212 f. 

,M Pappus is not alone in proposing the theoretic form of analysis; there arc com¬ 
parable statements from Proclus (cf. In Euclidem , p. 42; if indeed the "analytic faculty," 
or analytike dynamis . mentioned there refers to the geometric analysis) and from his 
disciple Marimis (In Euclidis Data). The latter writes, for instance, that 

the knowledge of the data is most necessary for the puqwses of the so-called analytic corpus 
doptn anatyomenos). .analysis is the finding of proof, it is compiled for us for linding the 
picHif of ihc similar things, and acquiring the analytic faculty is greater than having many 
proofs of tilings ill part. (Mcngc. Euclidis Opera, VI, pp. 252-254) 

In citing these passages, Hultsch notes their resemblance to Pappus' account of analysis, 
the verbal reminiscences hardly assignable to mere coincidence ( Collection , III, pp. 1275 
0. Hultsch infers, not the later writers 1 use of Pappus, but their common dependence on 
a much older account of analysis. Since Euclid, Apollonius, and Aristacus were the 
principal authors of this corpus, lie reasons, one of them must have been responsible for 
this alleged account, and doubtless Euclid himself was the one, since he was the oldest 
among them. Apparently, one must suppose that geometers could not apply the method 


Appraisal of the Analytic Field in Antiquity 379 


of analysis without also preparing and handing down a form of philosophical statement 
on its nature; and apparently, a later commentator like Gcminus was incapable of com¬ 
posing such a statement without an explicit model in hand deriving ultimately from Euclid 
or his disciples. 

"* Sec my account of this theorem in Evolution of the Euclidean Elements , Ch. 
Vtll/IV, especially pp. 279 If. Proposition XIII, I is also used in the proof of XIII, 6; 
hut the latter has been convincingly argued to be an i met point ion (cf. Heath. Euclid's 
Elements , III, p. 451). 

M,> liven with this modification, the analysis in this case would be unnecessary, 
since the claimed result is an obvious feature of the construction of this section of the 
line in Elements , II. II. That is, x is produced by taking the line x + z equal to the 
hypotenuse of a right triangle whose legs are z and 2z (~ x \ >•); see Chapter 3. One 
is thus led to suspect (hat XIII, I, like XIII, 6, is an interpolation—whence perhaps the 
entire series til Lemmas 1**6 may \k a later addition to Hie body of constructions and 
theorems of that hook. 

,n * Proclus, In Euclidem , p. 242. See note 67 for other references to Carpus. 

,,M Proclus. ibid For further references to this debate, see notes 60ff. 

One may cite Eutociiis* removal of the analysis in Itis version of the hyperbola 
construction (sec Chapter 7, note 33) and Pappus' stated preference for the synthetic 
mode by virtue of its greater conciseness in comparison with the analytic mode ( Collection, 
I, pp. 410-412). One notes that the angle trisections presented by Thabit ibn Qurra and 
Ahmad ihn Musa tscc my "Transmission of Geometry from Greek into Arabic") eliminate 
the analyses Pappus presents (cf. Collection , I, pp. 270 ff). Other examples of the 
commentators* preference for syntheses are cited in note 89 above. 

Collection (VII), II, pp 650*652. 

An alternative translation: "only that this is what is sought," seems possible, 
hut docs not conform to (lie sense of the passage. 

Collection fill), I, p. 30. 

1,4 Ibid., (IV). I, p. 284. 

For references to the accounts by ps.* Alexander and by Alexander as reported 
by Philoponus, see Chapter 3. 

"* Philoponus, In Analvtica Posterior . ed. M. Wallies (CAG 13). p. 112; sec 
Chapter 3. 

,M Commentary on Dimension of the Circle , Prop. I, in Archimedes, Opera . III, 
p. 230. 

"* See Chapter 3. 

M ’/n Categorias , cd. A. Btisse (CAG 4, Pt. 4, 1895), p. 75. 

"" Philoponus, op. c/7., p. 120; Simplicius, In Categorias, ed. K. Kalbflcisch (CAG 
K, 1907), p. 192. 

1,1 Cf. Simplicius, lor. cit. and also In Physica , cd. Diels, p. 60. 

Commentary on the Data in Euclid, Opera VI, ed. Mcngc, p, 240. 

1,1 See the survey in my Evolution of the Euclidean Elements , c.g., Ch. IX. 

1,4 Opera III, ed. Schbne, p. 172; for an account, sec Heath, History, II, pp. 339 
f. Hem s construction here seems related to another problem in the sectioning of a circle 
in Euclid** Division of Figures , Prop. 29 (cf. Heath, ibid.. I, p. 429). Note that in holh 







380 Ancient Tradition of Geometric Problems 


Appraisal of the Analytic Field in Antiquity 381 


cases, although the problem seeks a part which is one-third the circle, the method is 
clearly a general one. 

'Phis conjecture is made by Muimonidcs in his comment to Misfwah Eruvin I, 
5 (for which reference I thank Daniel Ixhtuann, Department of Mathematics, University 
of Southern California). Maurolico bases his own conjecture on the isopeiimciric property 
of the circle; cf. M. Clagctt, Archimedes in the Middle Aftes, III, pp. 779n, 7H2n. For 
references to Scifcl amt Gregory, see Hobson, op. nr. p. 31. Note that Descartes seems 
to be of the same view when he observes that the motions which yield the spiral, the 
quadratrix, and other such curves, that is, a circular motion synchronized with a linear 
motion, do not have a ratio which is exactly measurable (Geometrie. II, cd. Smith, p. 
45). He may well have in mind the similar remark made by Sparus, as reported by Pappus 
(Collection, I, p. 254), (hat the synchronization requires that this ratio be known, yet the 
object of the use of these curves is toward rinding that ratio. On Lambert's proof of the 
irrationality of it (1761/68), see Hobson, op. cit., pp. 43 ff; and Rnriques, op. cit., art. 
8 . 

Lulocius. in Archimedes, Opera, 111, p. 260. 

1,1 Ibid., p. 232. 

hi Physica VII.4, cd. Diels, pp. 1082 f. Simplicius also observes here that the 
circle quadrature "has not yet been found; but even if it might seem at this time to have 
been found, it is only through certain disputed hypotheses." One seems to hear an echo 
of this view in a remark by the Banu Musa: that one can compute an approximation to 
the value of the ratio of the circumference and diameter as closely as desired via ihc 
given (Archimedean) method, while "no one has to this day found anything beyond what 
has appeared to us" (Verba ftliorum, Prop. 6; Clagclt, Archimedes , I, p. 264). 

lf * hi Physica. p. 60. 

In Catefforias, p. 192. 

111 Collection (IV), I, pp. 252 ff; cf. note 125 above. 

1,1 Cf. the survey of angle trisect ions by al-Sijzi (cited in Woepeke, Omar , p. 120), 
where ihc "mobile" manner of the "ancient" (i.e., Archimedean) construction is crit¬ 
icized. Most forms of the Arabic terms, thaht ("fixed") and bayyin ("manifest"), arc 
indistinguishable in unpointed text. In the account of Nicomedes' cube duplication (see 
chapter 6) by Abu in"far Muh. b. al-Husain it is the ancient "method of instruments" 
which is replaced by a "geometric" procedure, the latter being the same as Pappus' use 
of conics for effecting the neusis for the angle trisection; cf. Sezgin, op. cit., p. 306. 

1,1 Cf. Vito Morcelli , xiv and the references cited in Chapier 2. 

See the citation by Pappus in Collection (VIII), III, p. 1026. 

•" See Chapter 6. 

m On the cube duplication, see the references in Chapter 5, note 115. Note his 
inclusion of practical methods for the measurement of figures in Metrica 1, 39 and II, 
20; his remarks on the practical interest of measures of the torus and of the sectioned 
cylinder (II, 13 and 16); his devotion of the entire writing On the Dioptra to the practical 
use of this angle-sighting device in surveying. 

,n Collection (VIII), 111, pp. 1072-74. Ilultsch's translation of this as proble- 
mata .. .sine demonstratione xcomctrica solvtmtio (that is, "problems solved without 
geometric demonstration"; ibid., p. 1075) cannot be right. 


See the remarks on this problem in Chapter 7. 

#n »* » » I*""* with considerable force by Descartes in the opening section 
of Bt>ok II id tils Geometric. 

M, \Soc. lor instance, liucliil's Hook XI, Dcf. IK and Apollonius' Book I, Dcf. I. 
In their comments on this part of Apollonius, neither Butociiis nor Pappus indicates 
knowledge of any ancient attempt to formalize these conceptions. 

On these neuses, see note 38 in Chapier 7. 

w See. respectively. Chapters 2(ii); and 5<ii); and 6 (note 106). 

For the text, see Philoponos, hi Ana . Post., p. 105n; reproduced by Heiberg in 
Apollonius, Opera , II, p 106. 

See Chapter 7. 

w Histories II. 2. 






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